Solved: Geometry The sides of a square are 27 inches in

An ________ ________ is a function whose domain is the set of positive integers.

A sequence is a ________ sequence when the domain of the function consists only of the first positive integers.

If you are given one or more of the first few terms of a sequence, and all other terms of the sequence are defined using previous terms, then the sequence is said to be defined ________.

If is a positive integer, then ________ is defined as n! 1 2 3 4 . . . n n n 1n.n

For the sum is called the ________ of summation, is the ________ limit of summation, and 1 is the ________ limit of summation.

The sum of the terms of a finite or infinite sequence is called a ________.

In Exercises 722, write the first five terms of the sequence. (Assume that begins with 1.) an 4n 7

In Exercises 722, write the first five terms of the sequence. (Assume that begins with 1.) an 2 1 3n

In Exercises 722, write the first five terms of the sequence. (Assume that begins with 1.) an 2n a

In Exercises 722, write the first five terms of the sequence. (Assume that begins with 1.) an 1 2 n an

In Exercises 722, write the first five terms of the sequence. (Assume that begins with 1.) an n n 2

In Exercises 722, write the first five terms of the sequence. (Assume that begins with 1.) an 6n 3n2 1

In Exercises 722, write the first five terms of the sequence. (Assume that begins with 1.) an 1 1n n an

In Exercises 722, write the first five terms of the sequence. (Assume that begins with 1.) an 1n n2 an

In Exercises 722, write the first five terms of the sequence. (Assume that begins with 1.) an 2n 3n

In Exercises 722, write the first five terms of the sequence. (Assume that begins with 1.) an 1 n32

In Exercises 722, write the first five terms of the sequence. (Assume that begins with 1.) an 2 3

In Exercises 722, write the first five terms of the sequence. (Assume that begins with 1.) an 1 1n an

In Exercises 722, write the first five terms of the sequence. (Assume that begins with 1.) a 2 6n nn 1n 2 an 1

In Exercises 722, write the first five terms of the sequence. (Assume that begins with 1.) an nn a 2 6n

In Exercises 722, write the first five terms of the sequence. (Assume that begins with 1.) an 1n n n 1 a

In Exercises 722, write the first five terms of the sequence. (Assume that begins with 1.) an 1n1 n2 1 an 

In Exercises 2326, find the indicated term of the sequence. a nn 1n 1n3n 2 an  a25 a

In Exercises 2326, find the indicated term of the sequence. an 1n1 a nn 1n 1n a16

In Exercises 2326, find the indicated term of the sequence. an 4n 2n2 3 a11 a

In Exercises 2326, find the indicated term of the sequence. an 4n2 n 3 nn 1n 2an 4n a13

In Exercises 2732, use a graphing utility to graph the first 10 terms of the sequence. (Assume that begins with 1.) an 2 3 n

In Exercises 2732, use a graphing utility to graph the first 10 terms of the sequence. (Assume that begins with 1.) an 2 4 n

In Exercises 2732, use a graphing utility to graph the first 10 terms of the sequence. (Assume that begins with 1.) an 160.5n1 a

In Exercises 2732, use a graphing utility to graph the first 10 terms of the sequence. (Assume that begins with 1.) an 80.75n1 an

In Exercises 2732, use a graphing utility to graph the first 10 terms of the sequence. (Assume that begins with 1.) an 2n n 1

In Exercises 2732, use a graphing utility to graph the first 10 terms of the sequence. (Assume that begins with 1.) an 3n2 n2 1 a

In Exercises 3336, match the sequence with the graph of its first 10 terms. [The graphs are labeled (a), (b), (c), and (d).] an 8 n 1

In Exercises 3336, match the sequence with the graph of its first 10 terms. [The graphs are labeled (a), (b), (c), and (d).] an 8n n 1 a

In Exercises 3336, match the sequence with the graph of its first 10 terms. [The graphs are labeled (a), (b), (c), and (d).] an 40.5n1 a

In Exercises 3336, match the sequence with the graph of its first 10 terms. [The graphs are labeled (a), (b), (c), and (d).] an 4n n!

In Exercises 3748, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) 3, 7, 11, 15, 19, . . .

In Exercises 3748, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) 0, 3, 8, 15, 24, . . .

In Exercises 3748, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) , . . . 2 3, 3 4, 4 5, 5 6, 6 7

In Exercises 3748, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) 2, 1 4, 1 8, 1 16 , . . .

In Exercises 3748, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) 2 1, 3 3, 4 5, 5 7, 6 9,

In Exercises 3748, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) 3, 2 9, 4 27, 8 81

In Exercises 3748, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) 1, , . . . 1 4, 1 9, 1 16, 1 25

In Exercises 3748, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) 1, 1 2, 1 6, 1 24, 1 120

In Exercises 3748, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) 1, 1, 1, 1, 1, .

In Exercises 3748, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) 1, 3, 1, 3, 1,

In Exercises 3748, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) 1, 3, 32 2 , 33 6 , 34 24, 35 120,

In Exercises 3748, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) 1 1 2, 1 3 4, 1 7 8, 1 15 16, 1 31 32, . . .

In Exercises 4952, write the first five terms of the sequence defined recursively. ak1 a a k 4 1 28, 97

In Exercises 4952, write the first five terms of the sequence defined recursively. ak1 2a a k 11 3, ak1

In Exercises 4952, write the first five terms of the sequence defined recursively. a0 1, a1 2, ak ak2 1 2ak2ak1

In Exercises 4952, write the first five terms of the sequence defined recursively. a0 1, a1 1, ak ak2 ak1 a0

In Exercises 5356, write the first five terms of the sequence defined recursively. Use the pattern to write the th term of the sequence as a function of n a1 6, ak1 ak 2 n n

In Exercises 5356, write the first five terms of the sequence defined recursively. Use the pattern to write the th term of the sequence as a function of n a1 25, ak1 ak 5 a1

In Exercises 5356, write the first five terms of the sequence defined recursively. Use the pattern to write the th term of the sequence as a function of n a1 81, 3ak

In Exercises 5356, write the first five terms of the sequence defined recursively. Use the pattern to write the th term of the sequence as a function of n a1 14, ak1 2ak ak1

In Exercises 57 and 58, use the Fibonacci sequence. (See Example 5.) Write the first 12 terms of the Fibonacci sequence and the first 10 terms of the sequence given by b n 1. n an1 an ,

In Exercises 57 and 58, use the Fibonacci sequence. (See Example 5.) Using the definition for in Exercise 57, show that can be defined recursively by bn 1 1 bn1 .

In Exercises 5962, write the first five terms of the sequence. (Assume that begins with 0.) an 5 n!

In Exercises 5962, write the first five terms of the sequence. (Assume that begins with 0.) an n! 2n 1 a

In Exercises 5962, write the first five terms of the sequence. (Assume that begins with 0.) an 1 n 1! an

In Exercises 5962, write the first five terms of the sequence. (Assume that begins with 0.) an 12n1 2n 1! an 1

In Exercises 6366, simplify the factorial expression. 4! 6!

In Exercises 6366, simplify the factorial expression. 12! 4! 8!

In Exercises 6366, simplify the factorial expression. n 1! n! 1

In Exercises 6366, simplify the factorial expression. 2n 1! 2n 1! n

In Exercises 6774, find the sum. 2 3 5 i1 2i 1 2n

In Exercises 6774, find the sum.  5 j3 1 j 2 3

In Exercises 6774, find the sum.  4 k1 10

In Exercises 6774, find the sum.  4 i0 i 2

In Exercises 6774, find the sum.  5 k2 k 12k 3  4

In Exercises 6774, find the sum. 4 i1 i 12 i 13  5 k2

In Exercises 6774, find the sum.  4 i1 2i

In Exercises 6774, find the sum.  4 j0 2j

In Exercises 7578, use a graphing utility to find the sum. 5 n0 1 2n 1

In Exercises 7578, use a graphing utility to find the sum.  4 k0 1k k 1 5

In Exercises 7578, use a graphing utility to find the sum.4 k0 1k k! 

In Exercises 7578, use a graphing utility to find the sum. 25 n0 1 4 n

In Exercises 7988, use sigma notation to write the sum. 1 31 1 32 1 33 . . . 1 39  25 n0

In Exercises 7988, use sigma notation to write the sum. 5 1 1 5 1 2 5 1 3 . . . 5 1 15 1 31

In Exercises 7988, use sigma notation to write the sum. 21 8322 83. . . 28 83 5 1 1

In Exercises 7988, use sigma notation to write the sum. 1 1 6 2 1 2 6 2 . . . 1 6 6 2  21 83

In Exercises 7988, use sigma notation to write the sum. 3 9 27 81 243 729

In Exercises 7988, use sigma notation to write the sum. 1 1 2 1 4 1 8 . . . 1 128

In Exercises 7988, use sigma notation to write the sum. 12 1 22 1 32 1 42 . . . 1 202

In Exercises 7988, use sigma notation to write the sum. 1 3 1 2 4 1 3 5 . . . 1 10 12 1

In Exercises 7988, use sigma notation to write the sum. 1 4 3 8 7 16 15 32 31 64 1

In Exercises 7988, use sigma notation to write the sum. 1 2 2 4 6 8 24 16 120 32 720 64 1

In Exercises 8992, find the indicated partial sum of the series.  i1 5 1 2 i Fourth partial sum

In Exercises 8992, find the indicated partial sum of the series.  i1 2 1 3 i  Fifth partial sum

In Exercises 8992, find the indicated partial sum of the series. n1 41 2 n  Third partial sum

In Exercises 8992, find the indicated partial sum of the series.  n1 81 4 n  Fourth partial sum

In Exercises 9396, find the sum of the infinite series.  i1 6 10i

In Exercises 9396, find the sum of the infinite series.  k1  1 10

In Exercises 9396, find the sum of the infinite series.  k1 7 1 10 k

In Exercises 9396, find the sum of the infinite series.  i1 2 10 i

Compound Interest An investor deposits $10,000 in an account that earns 3.5% interest compounded quarterly. The balance in the account after quarters is given by (a) Write the first eight terms of the sequence. (b) Find the balance in the account after 10 years by computing the 40th term of the sequence. (c) Is the balance after 20 years twice the balance after 10 years? Explain.

The numbers (in thousands) of AIDS cases reported from 2003 through 2010 can be approximated by where is the year, with corresponding to 2003. (Source: U.S. Centers for Disease Control and Prevention) (a) Write the terms of this finite sequence. Use a graphing utility to construct a bar graph that represents the sequence. (b) What does the graph in part (a) say about reported cases of AIDS?

In Exercises 99 and 100, determine whether the statement is true or false. Justify your answer.  4 i1 i 2 2i 4 i1 i 2 2 4 i1 i An 10,

In Exercises 99 and 100, determine whether the statement is true or false. Justify your answer. 4 j1 2 j  6 j3 2 j2 

In Exercises 101103, use the following definition of the arithmetic mean of a set of measurements . . . , xn x . 3 x , 2 x , 1, Find the arithmetic mean of the six checking account balances $327.15, $785.69, $433.04, $265.38, $604.12, and $590.30. Use the statistical capabilities of a graphing utility to verify your result.

In Exercises 101103, use the following definition of the arithmetic mean of a set of measurements . . . , xn x . 3 x , 2 x , 1, Proof Prove that n i1 xi x0. x .

In Exercises 101103, use the following definition of the arithmetic mean of a set of measurements . . . , xn x . 3 x , 2 x , 1, Proof Prove that n i1 xi x2  n i1 x 2 i 1 n n i1 xi 2 .

HOW DO YOU SEE IT? The graph represents the first 10 terms of a sequence. Complete each expression for the apparent th term of the sequence. Which expressions are appropriate to represent the cost to buy MP3 songs at a cost of $1 per song? Explain. (a) (b) (c) an  n k1 an ! n 1! 2 4 6 8 10 2 4 6 8 10 n a a n n 1  n an

In Exercises 105 and 106, find the first five terms of the sequence. an xn n!

In Exercises 105 and 106, find the first five terms of the sequence. an 1n x2n1 2n 1 an x

Cube A cube is made up of 27 unit cubes (a unit cube has a length, width, and height of 1 unit), and only the faces of each cube that are visible are painted blue, as shown in the figure. (a) Complete the table to determine how many unit cubes of the cube have 0 blue faces, 1 blue face, 2 blue faces, and 3 blue faces. (b) Repeat part (a) for a cube, a cube, and a cube. (c) What type of pattern do you observe? (d) Write formulas you could use to repeat part (a) for an cube.

A sequence is called an ________ sequence when the differences between consecutive terms are the same. This difference is called the ________ difference.

The nth term of an arithmetic sequence has the form ________.

When you know the term of an arithmetic sequence and you know the common difference of the sequence, you can find the term by using the ________ formula n 1th an1 an d. nt

You can use the formula to find the sum of the first terms of an arithmetic sequence, which is called the ________ of a ________ ________ ________.

In Exercises 512, determine whether the sequence is arithmetic. If so, then find the common difference. 10, 8, 6, 4, 2, .

In Exercises 512, determine whether the sequence is arithmetic. If so, then find the common difference. 4, 9, 14, 19, 24, . . .

In Exercises 512, determine whether the sequence is arithmetic. If so, then find the common difference. 1, 2, 4, 8, 16, . . .

In Exercises 512, determine whether the sequence is arithmetic. If so, then find the common difference. 80, 40, 20, 10, 5,

In Exercises 512, determine whether the sequence is arithmetic. If so, then find the common difference. 9 4, 2, 7 4, 3 2, 5 4, . . .

In Exercises 512, determine whether the sequence is arithmetic. If so, then find the common difference. 5.3, 5.7, 6.1, 6.5, 6.9, . . .

In Exercises 512, determine whether the sequence is arithmetic. If so, then find the common difference. ln 1, ln 2, ln 3, ln 4, ln 5, . .

In Exercises 512, determine whether the sequence is arithmetic. If so, then find the common difference. 12, 22, 32, 42, 52, . .

In Exercises 1320, write the first five terms of the sequence. Determine whether the sequence is arithmetic. If so, then find the common difference. (Assume that begins with 1.) an 5 3n

In Exercises 1320, write the first five terms of the sequence. Determine whether the sequence is arithmetic. If so, then find the common difference. (Assume that begins with 1.) an 100 3n

In Exercises 1320, write the first five terms of the sequence. Determine whether the sequence is arithmetic. If so, then find the common difference. (Assume that begins with 1.) an 3 4n 2an 1

In Exercises 1320, write the first five terms of the sequence. Determine whether the sequence is arithmetic. If so, then find the common difference. (Assume that begins with 1.) an 1 n 14an

In Exercises 1320, write the first five terms of the sequence. Determine whether the sequence is arithmetic. If so, then find the common difference. (Assume that begins with 1.)an 1n an

In Exercises 1320, write the first five terms of the sequence. Determine whether the sequence is arithmetic. If so, then find the common difference. (Assume that begins with 1.) an 2n1 a

In Exercises 1320, write the first five terms of the sequence. Determine whether the sequence is arithmetic. If so, then find the common difference. (Assume that begins with 1.) an n 1n3 n an

In Exercises 1320, write the first five terms of the sequence. Determine whether the sequence is arithmetic. If so, then find the common difference. (Assume that begins with 1.) an 2n an n 1n

In Exercises 2130, find a formula for for the arithmetic sequence. a1 1, d 3 a

In Exercises 2130, find a formula for for the arithmetic sequence. a1 15, d 4

In Exercises 2130, find a formula for for the arithmetic sequence. a1 100, d 8 3

In Exercises 2130, find a formula for for the arithmetic sequence. a1 0, d 2 a

In Exercises 2130, find a formula for for the arithmetic sequence. 4, 10, 5, 0, 5, 10, . . . 3 2, 1, 7 2 , . . .

In Exercises 2130, find a formula for for the arithmetic sequence. 10, 5, 0, 5, 10, . .

In Exercises 2130, find a formula for for the arithmetic sequence. a1 5, a4 15 a

In Exercises 2130, find a formula for for the arithmetic sequence. a1 4, a5 16 4

In Exercises 2130, find a formula for for the arithmetic sequence. a3 94, a6 85 a

In Exercises 2130, find a formula for for the arithmetic sequence. a5 190, a10 115

In Exercises 3138, write the first five terms of the arithmetic sequence. a1 5, d 6 4

In Exercises 3138, write the first five terms of the arithmetic sequence. a1 5, d 3 a1

In Exercises 3138, write the first five terms of the arithmetic sequence. a1 a1 16.5, d 0.25 13 5 , d 2 5 a1 5

In Exercises 3138, write the first five terms of the arithmetic sequence. a1 16.5, d 0.25 1

In Exercises 3138, write the first five terms of the arithmetic sequence. a1 2, a12 46 a

In Exercises 3138, write the first five terms of the arithmetic sequence. a4 16, a10 46

In Exercises 3138, write the first five terms of the arithmetic sequence.. a8 26, a12 42 a

In Exercises 3138, write the first five terms of the arithmetic sequence. a3 19, a15 1.7a

In Exercises 3942, write the first five terms of the arithmetic sequence defined recursively a1 15, an1 an 4a

In Exercises 3942, write the first five terms of the arithmetic sequence defined recursively a1 200, an1 an 10 a1

In Exercises 3942, write the first five terms of the arithmetic sequence defined recursively a1 5 8, an1 an 18a

In Exercises 3942, write the first five terms of the arithmetic sequence defined recursively a1 0.375, an1 an 0.25

In Exercises 4346, the first two terms of the arithmetic sequence are given. Find the missing term. a1 5, a2 11, a10 a1

In Exercises 4346, the first two terms of the arithmetic sequence are given. Find the missing term. a1 3, a2 13, a9 a1

In Exercises 4346, the first two terms of the arithmetic sequence are given. Find the missing term. a1 4.2, a2 6.6, a7 a1

In Exercises 4346, the first two terms of the arithmetic sequence are given. Find the missing term. a1 0.7, a2 13.8, a8 a1 4.

In Exercises 4752, find the sum of the finite arithmetic sequence. 2 4 6 8 10 12 14 16 18 20

In Exercises 4752, find the sum of the finite arithmetic sequence. 1 4 7 10 13 16 19

In Exercises 4752, find the sum of the finite arithmetic sequence. 1 3579 1 4 7 10

In Exercises 4752, find the sum of the finite arithmetic sequence. 5 311 3 5 1 35

In Exercises 4752, find the sum of the finite arithmetic sequence. Sum of the first 100 positive odd integers

In Exercises 4752, find the sum of the finite arithmetic sequence. Sum of the integers from to 30

In Exercises 5358, find the indicated th partial sum of the arithmetic sequence. 8, 20, 32, 44, . . . , n 10

In Exercises 5358, find the indicated th partial sum of the arithmetic sequence. 6, 2, 2, 6, . . . , n 50 8,

In Exercises 5358, find the indicated th partial sum of the arithmetic sequence. 4.2, 3.7, 3.2, 2.7 . . . , n 12

In Exercises 5358, find the indicated th partial sum of the arithmetic sequence. 75, 70, 65, 60 . . . , n 25

In Exercises 5358, find the indicated th partial sum of the arithmetic sequence. a1 100, a25 220, n 25.

In Exercises 5358, find the indicated th partial sum of the arithmetic sequence. a n 100 1 15, a100 307,a1

In Exercises 5964, find the partial sum. 50 n1 n

In Exercises 5964, find the partial sum. 100 n51 7n

In Exercises 5964, find the partial sum. 30 n11 n  10 n1 n

In Exercises 5964, find the partial sum. 100 n51 n  50 n1 n 30

In Exercises 5964, find the partial sum. 500 n1 n 8

In Exercises 5964, find the partial sum. 250 n1 1000 n 50

In Exercises 6568, match the arithmetic sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] an an 3n 5 3 4 n 8 2

In Exercises 6568, match the arithmetic sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] an 3n 5 3

In Exercises 6568, match the arithmetic sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] an 2 an 25 3n 3 4 n a

In Exercises 6568, match the arithmetic sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] an 25 3n 3

In Exercises 6972, use a graphing utility to graph the first 10 terms of the sequence. (Assume that begins with 1.) an 15 3 2

In Exercises 6972, use a graphing utility to graph the first 10 terms of the sequence. (Assume that begins with 1.) an 5 2n a

In Exercises 6972, use a graphing utility to graph the first 10 terms of the sequence. (Assume that begins with 1.) an 0.2n 3

In Exercises 6972, use a graphing utility to graph the first 10 terms of the sequence. (Assume that begins with 1.) an 0.3n 8 a

In Exercises 7376, use a graphing utility to find the partial sum. 50 n0 50 2n a

In Exercises 7376, use a graphing utility to find the partial sum. 100 n1 n 1 2

In Exercises 7376, use a graphing utility to find the partial sum. 60 i1 250 2 5i 

In Exercises 7376, use a graphing utility to find the partial sum. 200 j1 10.5 0.025j

Seating Capacity Determine the seating capacity of an auditorium with 36 rows of seats when there are 15 seats in the first row, 18 seats in the second row, 21 seats in the third row, and so on.

Brick Pattern A triangular brick wall is made by cutting some bricks in half to use in the first column of every other row (see figure). The wall has 28 rows. The top row is one-half brick wide and the bottom row is 14 bricks wide. How many bricks are in the finished wall?

In Exercises 79 and 80, consider a job offer with the given starting salary and the given annual raise. (a) Determine the salary during the sixth year of employment. (b) Determine the total compensation from the company through six full years of employment. $32,500 $1500

In Exercises 79 and 80, consider a job offer with the given starting salary and the given annual raise. (a) Determine the salary during the sixth year of employment. (b) Determine the total compensation from the company through six full years of employment. $36,800 $1750

An object with negligible air resistance is dropped from the top of the Willis Tower in Chicago at a height of 1450 feet. During the first second of fall, the object falls 16 feet; during the second second, it falls 48 feet; during the third second, it falls 80 feet; during the fourth second, it falls 112 feet. If this arithmetic pattern continues, then how many feet will the object fall in 7 seconds?

Prize Money A county fair is holding a baked goods competition in which the top eight bakers receive cash prizes. First place receives a cash prize of $200, second place receives $175, third place receives $150, and so on. (a) Write a sequence that represents the cash prize awarded in terms of the place in which the baked good places. (b) Find the total amount of prize money awarded at the competition

Total Sales An entrepreneur sells $15,000 worth of sports memorabilia during one year and sets a goal of increasing annual sales by $5000 each year for 9 years. Assuming that the entrepreneur meets this goal, find the total sales during the first 10 years of this business. What kinds of economic factors could prevent the business from meeting its goals?

Borrowing Money You borrow $5000 from your parents to purchase a used car. The arrangements of the loan are such that you make payments of $250 per month plus 1% interest on the unpaid balance. (a) Find the first years monthly payments and the unpaid balance after each month. (b) Find the total amount of interest paid over the term of the loan

Data Analysis: Sales The table shows the sales (in billions of dollars) for Eli Lilly and Co. from 2004 through 2011. (Source: Eli Lilly and Co.) (a) Construct a bar graph showing the annual sales from 2004 through 2011. (b) Find an arithmetic sequence that models the data. Let represent the year, with corresponding to 2004. (c) Use a graphing utility to graph the terms of the finite sequence you found in part (a). (d) Use summation notation to represent the total sales from 2004 through 2011. Find the total sales.

Writing Explain how to use the first two terms of an arithmetic sequence to find the th term.

In Exercises 87 and 88, determine whether the statement is true or false. Justify your answer. Given an arithmetic sequence for which only the first two terms are known, it is possible to find the th term.

In Exercises 87 and 88, determine whether the statement is true or false. Justify your answer. If the only known information about a finite arithmetic sequence is its first term and its last term, then it is possible to find the sum of the sequence.

In Exercises 89 and 90, find the first 10 terms of the sequence. a a1 y, d 5y 1 x, d 2xn

In Exercises 89 and 90, find the first 10 terms of the sequence. a1 y, d 5y 1

Comparing Graphs of a Sequence and a Line (a) Graph the first 10 terms of the arithmetic sequence (b) Graph the equation of the line (c) Discuss any differences between the graph of and the graph of (d) Compare the slope of the line in part (b) with the common difference of the sequence in part (a). What can you conclude about the slope of a line and the common difference of an arithmetic sequence?

HOW DO YOU SEE IT? A steel ball with negligible air resistance is dropped from a plane. The figure shows the distance that the ball falls during each of the first four seconds after it is dropped. (a) Describe a pattern of the distances shown. Explain why the distances form a finite arithmetic sequence. (b) Assume the pattern described in part (a) continues. Describe the steps and formulas for using the sum of a finite sequence to find the total distance the ball falls in a given whole number of seconds n.

Pattern Recognition (a) Compute the following sums of consecutive positive odd integers. (b) Use the sums in part (a) to make a conjecture about the sums of consecutive positive odd integers. Check your conjecture for the sum (c) Verify your conjecture algebraically. Project: Municipal Waste To work an extended application analyzing the amounts of municipal waste recovered in the United States from 1991 through 2010, visit this texts website at LarsonPrecalculus.com. (Source: U.S. Environmental Protection Agency

A sequence is called a ________ sequence when the ratios of consecutive terms are the same. This ratio is called the ________ ratio.

The th term of a geometric sequence has the form ________.

The sum of a finite geometric sequence with common ratio is given by ________.

The sum of the terms of an infinite geometric sequence is called a ________ ________.

In Exercises 512, determine whether the sequence is geometric. If so, then find the common ratio. 2, 10, 50, 250

In Exercises 512, determine whether the sequence is geometric. If so, then find the common ratio. 3, 12, 21, 30, . .

In Exercises 512, determine whether the sequence is geometric. If so, then find the common ratio. 1 8, 1 4, 1 2, 1,

In Exercises 512, determine whether the sequence is geometric. If so, then find the common ratio. 9, 6, 4, 8 3,

In Exercises 512, determine whether the sequence is geometric. If so, then find the common ratio. 1, 5, 1, 0.2, 0.04, . . . 1 2, 1 3, 1 4

In Exercises 512, determine whether the sequence is geometric. If so, then find the common ratio. 5, 1, 0.2, 0.04,

In Exercises 512, determine whether the sequence is geometric. If so, then find the common ratio. 1, 7, 7, 77,

In Exercises 512, determine whether the sequence is geometric. If so, then find the common ratio. 2, 4 3 , 8 3 , 16 33

In Exercises 1322, write the first five terms of the geometric sequence. a1 4, r 3 a1

In Exercises 1322, write the first five terms of the geometric sequence. a1 8, r 2

In Exercises 1322, write the first five terms of the geometric sequence. a1 1, 4 r 1 2

In Exercises 1322, write the first five terms of the geometric sequence. a1 6, r 1 a1 1, 4 r 1

In Exercises 1322, write the first five terms of the geometric sequence. a1 1, r e a

In Exercises 1322, write the first five terms of the geometric sequence. a1 2, r

In Exercises 1322, write the first five terms of the geometric sequence. a1 3, r 5

In Exercises 1322, write the first five terms of the geometric sequence. a1 4, r 1 2 a

In Exercises 1322, write the first five terms of the geometric sequence. a a1 5, r 2x 1 2, r x 4 a1

In Exercises 1322, write the first five terms of the geometric sequence. a1 5, r 2x 1

In Exercises 2328, write the first five terms of the geometric sequence. Determine the common ratio and write the th term of the sequence as a function of n a1 64, ak1 3ak 1 2ak

In Exercises 2328, write the first five terms of the geometric sequence. Determine the common ratio and write the th term of the sequence as a function of n a1 81, ak1 1 a1 64, ak1 3ak 1

In Exercises 2328, write the first five terms of the geometric sequence. Determine the common ratio and write the th term of the sequence as a function of n a1 9, ak1 2ak a

In Exercises 2328, write the first five terms of the geometric sequence. Determine the common ratio and write the th term of the sequence as a function of n a1 5, ak1 2ak

In Exercises 2328, write the first five terms of the geometric sequence. Determine the common ratio and write the th term of the sequence as a function of n a1 6, ak1 2 ak 3 2ak

In Exercises 2328, write the first five terms of the geometric sequence. Determine the common ratio and write the th term of the sequence as a function of n a1 80, ak1 1 a1 6, ak1 2 ak 3

In Exercises 2938, write an expression for the th term of the geometric sequence. Then find the indicated term. a1 4, r , n 8 1 2, n 10 n

In Exercises 2938, write an expression for the th term of the geometric sequence. Then find the indicated term. a1 5, r 7 2 a1 4, r , n 8 1 2

In Exercises 2938, write an expression for the th term of the geometric sequence. Then find the indicated term. a1 6, r , n 10 1 3, n 12 a1

In Exercises 2938, write an expression for the th term of the geometric sequence. Then find the indicated term. a1 64, r 1 4 a1 6, r , n 10 1 3,

In Exercises 2938, write an expression for the th term of the geometric sequence. Then find the indicated term. a1 100, r e , n 4 x , n 9 a1

In Exercises 2938, write an expression for the th term of the geometric sequence. Then find the indicated term. a1 1, r ex a1 100, r e , n 4 x ,

In Exercises 2938, write an expression for the th term of the geometric sequence. Then find the indicated term. a1 1, r 2, n 12 a1

In Exercises 2938, write an expression for the th term of the geometric sequence. Then find the indicated term. a1 1, r 3, n 8a

In Exercises 2938, write an expression for the th term of the geometric sequence. Then find the indicated term. a1 500, r 1.02, n 40 a1

In Exercises 2938, write an expression for the th term of the geometric sequence. Then find the indicated term. a1 1000, r 1.005, n 60 a1

In Exercises 3946, find the indicated term of the geometric sequence. 39. 9th term: 11, 33, 99, . . .

In Exercises 3946, find the indicated term of the geometric sequence. 40. 7th term: 3, 36, 432, . . .

In Exercises 3946, find the indicated term of the geometric sequence. 8th term 2, 1 8, 1 32, 1 128,

In Exercises 3946, find the indicated term of the geometric sequence. 42. 7th term: 8 5, 16 25, 32 125, 64 625,

In Exercises 3946, find the indicated term of the geometric sequence. 3rd term: a1 16, 4 a4 27

In Exercises 3946, find the indicated term of the geometric sequence. 1st term:a5 3 a2 3, 64 a

In Exercises 3946, find the indicated term of the geometric sequence. 6th term a4 18, 3 aa7 2

In Exercises 3946, find the indicated term of the geometric sequence. 5th term a2 2, a3 2 a7

In Exercises 4750, match the geometric sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] an 18 2 3 n1 2

In Exercises 4750, match the geometric sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] an 182 3 n1 an

In Exercises 4750, match the geometric sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] an 18 3 2 n1 an

In Exercises 4750, match the geometric sequence with its graph. [The graphs are labeled (a), (b), (c), and (d).] an 183 2 n1 an 1

In Exercises 5154, use a graphing utility to graph the first 10 terms of the sequence. an 101.5n1 an

In Exercises 5154, use a graphing utility to graph the first 10 terms of the sequence. an 120.4n1 an 1

In Exercises 5154, use a graphing utility to graph the first 10 terms of the sequence. an 201.25n1 an

In Exercises 5154, use a graphing utility to graph the first 10 terms of the sequence. an 21.3n1 an

In Exercises 5568, find the sum of the finite geometric sequence. n1 4n1

In Exercises 5568, find the sum of the finite geometric sequence.  10 n1 3 2 n1 

In Exercises 5568, find the sum of the finite geometric sequence.  6 n1 7n1 

In Exercises 5568, find the sum of the finite geometric sequence.  8 n1 55 2 n1  6

In Exercises 5568, find the sum of the finite geometric sequence.  7 i1 641 2 i1  8

In Exercises 5568, find the sum of the finite geometric sequence.  12 i1 16 1 2 i1 

In Exercises 5568, find the sum of the finite geometric sequence.  20 n0 3 3 2 n

In Exercises 5568, find the sum of the finite geometric sequence.  40 n0 5 3 5 n 

In Exercises 5568, find the sum of the finite geometric sequence.  20 n0 10 1 5 n 

In Exercises 5568, find the sum of the finite geometric sequence.  6 n0 5001.04n 

In Exercises 5568, find the sum of the finite geometric sequence.  40 n0 21 4 n 

In Exercises 5568, find the sum of the finite geometric sequence.  50 n0 10 2 3 n1 

In Exercises 5568, find the sum of the finite geometric sequence.  100 i1 15 2 3 i1 

In Exercises 5568, find the sum of the finite geometric sequence. 10 i1 81 4 i1  1

In Exercises 6972, use summation notation to write the sum. 10 30 90 ... 7290

In Exercises 6972, use summation notation to write the sum. 15 3 3 5 ... 3 625 1

In Exercises 6972, use summation notation to write the sum. 0.1 0.4 1.6 ... 102.4

In Exercises 6972, use summation notation to write the sum. 32 24 18 ... 10.125

In Exercises 7380, find the sum of the infinite geometric series.   n0 1 2 n 3

In Exercises 7380, find the sum of the infinite geometric series.   n0 1 2 n 

In Exercises 7380, find the sum of the infinite geometric series.  n0 22 3 n 

In Exercises 7380, find the sum of the infinite geometric series.   n0 40.2n 

In Exercises 7380, find the sum of the infinite geometric series. 8 6 9 2 27 8 .

In Exercises 7380, find the sum of the infinite geometric series. 9 6 4 8 3 .

In Exercises 7380, find the sum of the infinite geometric series. 9 1 3 1 3 .

In Exercises 7380, find the sum of the infinite geometric series. 125 36 25 6 5 6

In Exercises 81 and 82, find the rational number representation of the repeating decimal. 0.36

In Exercises 81 and 82, find the rational number representation of the repeating decimal. 0.318

In Exercises 83 and 84, use a graphing utility to graph the function. Identify the horizontal asymptote of the graph and determine its relationship to the sum. fx6 1 0.5x 1 0.5 , 0.36 0   n 0 6 1 2

In Exercises 83 and 84, use a graphing utility to graph the function. Identify the horizontal asymptote of the graph and determine its relationship to the sum. fx2 1 0.8x 1 0.8 ,   n   n 0 2 4 5 n

Depreciation A tool and die company buys a machine for $175,000 and it depreciates at a rate of 30% per year. (In other words, at the end of each year the depreciated value is 70% of what it was at the beginning of the year.) Find the depreciated value of the machine after 5 full years

The table shows the mid-year populations of China (in millions) from 2004 through 2010. (Source: U.S. Census Bureau) (a) Use the exponential regression feature of a graphing utility to find a geometric sequence that models the data. Let represent the year, with corresponding to 2004. (b) Use the sequence from part (a) to describe the rate at which the population of China is growing. (c) Use the sequence from part (a) to predict the population of China in 2017. The U.S. Census Bureau predicts the population of China will be 1372.1 million in 2017. How does this value compare with your prediction? (d) Use the sequence from part (a) to determine when the population of China will reach 1.4 billion.

Annuity An investor deposits dollars on the first day of each month in an account with an annual interest rate compounded monthly. The balance after years is Show that the balance is

Annuity An investor deposits $100 on the first day of each month in an account that pays 2% interest, compounded monthly. The balance in the account at the end of 5 years is Use the result of Exercise 87 to find A

In Exercises 89 and 90, use the following information. A state government gives property owners a tax rebate with the anticipation that each property owner will spend approximately of the rebate, and in turn each recipient of this amount will spend of what he or she receives, and so on. Economists refer to this exchange of money and its circulation within the economy as the multiplier effect. The multiplier effect operates on the idea that the expenditures of one individual become the income of another individual. For the given tax rebate, find the total amount put back into the states economy, assuming that this effect continues without end. 89. $400 75%

In Exercises 89 and 90, use the following information. A state government gives property owners a tax rebate with the anticipation that each property owner will spend approximately of the rebate, and in turn each recipient of this amount will spend of what he or she receives, and so on. Economists refer to this exchange of money and its circulation within the economy as the multiplier effect. The multiplier effect operates on the idea that the expenditures of one individual become the income of another individual. For the given tax rebate, find the total amount put back into the states economy, assuming that this effect continues without end. 90. $600 72.5%

Geometry The sides of a square are 27 inches in length. New squares are formed by dividing the original square into nine squares. The center square is then shaded (see figure). If this process is repeated three more times, then determine the total area of the shaded region.

Distance A ball is dropped from a height of 6 feet and begins bouncing as shown in the figure. The height of each bounce is three-fourths the height of the previous bounce. Find the total vertical distance the ball travels before coming to rest.

Salary An investment firm has a job opening with a salary of $45,000 for the first year. During the next 39 years, there is a 5% raise each year. Find the total compensation over the 40-year period.

HOW DO YOU SEE IT? Use the figures shown below. (i) (ii) (a) Without performing any calculations, determine which figure shows the terms of the sequence and which shows the terms of Explain your reasoning. (b) Which sequence has terms that can be summed? Explain your reasoning.

In Exercises 95 and 96, determine whether the statement is true or false. Justify your answer. A sequence is geometric when the ratios of consecutive differences of consecutive terms are the same.

In Exercises 95 and 96, determine whether the statement is true or false. Justify your answer. To find the th term of a geometric sequence, multiply its common ratio by the first term of the sequence raised to the th power

In Exercises 95 and 96, determine whether the statement is true or false. Justify your answer. Graphical Reasoning Consider the graph of (a) Use a graphing utility to graph for and What happens as (b) Use the graphing utility to graph for 2, and 3. What happens as

In Exercises 95 and 96, determine whether the statement is true or false. Justify your answer. Writing Write a brief paragraph explaining why the terms of a geometric sequence decrease in magnitude when Project: Housing Vacancies To work an extended application analyzing the numbers of vacant houses in the United States from 1990 through 2011, visit this texts website at LarsonPrecalculus.com. (Source: U.S. Census Bureau)

The first step in proving a formula by ________ ________ is to show that the formula is true when n 1.

To find the ________ differences of a sequence, subtract consecutive terms.

A sequence is an ________ sequence when the first differences are all the same nonzero number

If the ________ differences of a sequence are all the same nonzero number, then the sequence has a perfect quadratic model.

In Exercises 510, find for the given Pk. Pk 5 kk 1 Pk

In Exercises 510, find for the given Pk. Pk 1 2k 2Pk

In Exercises 510, find for the given Pk. P 2k 1k k 2k 32 6 Pk 1

In Exercises 510, find for the given Pk. Pk k 3 P 2k 1k k

In Exercises 510, find for the given Pk. Pk 3 k 2k 3 Pk k

In Exercises 510, find for the given Pk. Pk k2 2k 12 Pk

In Exercises 1124, use mathematical induction to prove the formula for every positive integer n 2 4 6 8 . . . 2n nn 1 n. Pk k

In Exercises 1124, use mathematical induction to prove the formula for every positive integer n 3 6 9 12 . . . 3n 3n 2 n 1 2 4 6

In Exercises 1124, use mathematical induction to prove the formula for every positive integer n 2 7 12 17 . . . 5n 3n 2 5n 1 3 6 9 12

In Exercises 1124, use mathematical induction to prove the formula for every positive integer n 1 4 7 10 . . . 3n 2n 2 3n 1 2 7 12

In Exercises 1124, use mathematical induction to prove the formula for every positive integer n 1 2 22 23 . . . 2n1 2n 1 1 4

In Exercises 1124, use mathematical induction to prove the formula for every positive integer n 21 3 32 33 . . . 3n13n 1 1 2 22

In Exercises 1124, use mathematical induction to prove the formula for every positive integer n 1 2 3 4 . . . n nn 1 2 21 3 32

In Exercises 1124, use mathematical induction to prove the formula for every positive integer n 13 23 33 43 . . . n3 n2n 12 4 1 2

In Exercises 1124, use mathematical induction to prove the formula for every positive integer n 12 32 52 . . . 2n 12 n2n 12n 1 3 13 23 33 43

In Exercises 1124, use mathematical induction to prove the formula for every positive integer n 1 1 1 1 1 2 1 1 3 . . . 1 1 n n 1 12 3

In Exercises 1124, use mathematical induction to prove the formula for every positive integer n n i1 i5 n2n 122n2 2n 1 12 1 1

In Exercises 1124, use mathematical induction to prove the formula for every positive integer n n i1 i 4 nn 12n 13n2 3n 1 30 n i1

In Exercises 1124, use mathematical induction to prove the formula for every positive integer n n i1 ii 1nn 1n 2 3 n i1

In Exercises 1124, use mathematical induction to prove the formula for every positive integer n n i1 1 2i 12i 1 n 2n 1 n

In Exercises 2530, use mathematical induction to prove the inequality for the indicated integer values of n n! > 2 > n, n 7 n, n 4

In Exercises 2530, use mathematical induction to prove the inequality for the indicated integer values of n 4 3 n n! > 2 > n, n 7 n

In Exercises 2530, use mathematical induction to prove the inequality for the indicated integer values of n 1 1 2 1 3 . . . 1 n > n, n 2 4

In Exercises 2530, use mathematical induction to prove the inequality for the indicated integer values of n  x y n1 <  x y n , n 1 0 < x < y

In Exercises 2530, use mathematical induction to prove the inequality for the indicated integer values of n 1 an 1 a > 0 n na, n

In Exercises 2530, use mathematical induction to prove the inequality for the indicated integer values of n 2n n 3 2 > n 12, 1

In Exercises 3140, use mathematical induction to prove the property for all positive integers n abn an bn

In Exercises 3140, use mathematical induction to prove the property for all positive integers n  a b n an bn

In Exercises 3140, use mathematical induction to prove the property for all positive integers n x1 0, x2 0, . . . , xn 0, x1 x 2 x3 . . . xn 1 x1 1x3 1 . . . xn

In Exercises 3140, use mathematical induction to prove the property for all positive integers n x1 > 0, x2 > 0, . . . , xn > 0, . . . ln xn lnx . 1 x2 .

In Exercises 3140, use mathematical induction to prove the property for all positive integers n 35. Generalized Distributive Law: xy1 y2 . . . yn xy1 xy2 . . . xyn . . . ln

In Exercises 3140, use mathematical induction to prove the property for all positive integers n a bin a bin xy1 are complex conjugates for all n 1.

In Exercises 3140, use mathematical induction to prove the property for all positive integers n A factor of is 3

In Exercises 3140, use mathematical induction to prove the property for all positive integers n A factor of is 2.

In Exercises 3140, use mathematical induction to prove the property for all positive integers n A factor of is 3

In Exercises 3140, use mathematical induction to prove the property for all positive integers n A factor of is 5.

In Exercises 4144, use mathematical induction to find a formula for the sum of the first terms of the sequence. 1, 5, 9, 13

In Exercises 4144, use mathematical induction to find a formula for the sum of the first terms of the sequence. 3, 9 2, 27 4 , 81 8 1, 5, 9, 13, . . . , . .

In Exercises 4144, use mathematical induction to find a formula for the sum of the first terms of the sequence. 1 4 , 1 12, 1 24, 1 40, . . . , 1 2nn 1 , .

In Exercises 4144, use mathematical induction to find a formula for the sum of the first terms of the sequence. 2 3 , 1 3 4 , 1 4 5 , 1 5 6 , n 1n 2 , . .

In Exercises 4554, find the sum using the formulas for the sums of powers of integers. n 15 n1 n

In Exercises 4554, find the sum using the formulas for the sums of powers of integers. 30 n1 n

In Exercises 4554, find the sum using the formulas for the sums of powers of integers. 6 n1 n2

In Exercises 4554, find the sum using the formulas for the sums of powers of integers. 10 n1 n3

In Exercises 4554, find the sum using the formulas for the sums of powers of integers. 5 n1 n4

In Exercises 4554, find the sum using the formulas for the sums of powers of integers. 8 n1 n5

In Exercises 4554, find the sum using the formulas for the sums of powers of integers. 3 n 6 n1 n2 n

In Exercises 4554, find the sum using the formulas for the sums of powers of integers. 20 n1 n 3 n

In Exercises 4554, find the sum using the formulas for the sums of powers of integers. 6 i1 6i 8i 3

In Exercises 4554, find the sum using the formulas for the sums of powers of integers. 10 j1 3 1 2 j 1 2 j 2

In Exercises 5560, decide whether the sequence can be represented perfectly by a linear or a quadratic model. If so, then find the model. 5, 13, 21, 29, 37, 45, . . .

In Exercises 5560, decide whether the sequence can be represented perfectly by a linear or a quadratic model. If so, then find the model. 2, 9, 16, 23, 30, 37, . . .

In Exercises 5560, decide whether the sequence can be represented perfectly by a linear or a quadratic model. If so, then find the model. 6, 15, 30, 51, 78, 111, . . .

In Exercises 5560, decide whether the sequence can be represented perfectly by a linear or a quadratic model. If so, then find the model. 0, 6, 16, 30, 48, 70, . . .

In Exercises 5560, decide whether the sequence can be represented perfectly by a linear or a quadratic model. If so, then find the model. 2, 1, 6, 13, 22, 33, . . .

In Exercises 5560, decide whether the sequence can be represented perfectly by a linear or a quadratic model. If so, then find the model. 1, 8, 23, 44, 71, 104, . . .

In Exercises 6168, write the first six terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a perfect linear model, a perfect quadratic model, or neither a1 0 an an1 3 an

In Exercises 6168, write the first six terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a perfect linear model, a perfect quadratic model, or neither a1 2 an an1 2

In Exercises 6168, write the first six terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a perfect linear model, a perfect quadratic model, or neither a1 3 a an an1 n

In Exercises 6168, write the first six terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a perfect linear model, a perfect quadratic model, or neither a2 3 an 2an1

In Exercises 6168, write the first six terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a perfect linear model, a perfect quadratic model, or neither a0 2 an an1an an1 n 2 a0

In Exercises 6168, write the first six terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a perfect linear model, a perfect quadratic model, or neither a0 0 an an1 n 2

In Exercises 6168, write the first six terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a perfect linear model, a perfect quadratic model, or neither a1 2 an n an1

In Exercises 6168, write the first six terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a perfect linear model, a perfect quadratic model, or neither a1 0 an an1 2n

In Exercises 6974, find a quadratic model for the sequence with the indicated terms. a0 3, a1 3, a4 15an

In Exercises 6974, find a quadratic model for the sequence with the indicated terms. a0 7, a1 6, a3 10 a0

In Exercises 6974, find a quadratic model for the sequence with the indicated terms. a0 3, a2 1, a4 9 a0

In Exercises 6974, find a quadratic model for the sequence with the indicated terms. a0 3, a2 0, a6 36 a0

In Exercises 6974, find a quadratic model for the sequence with the indicated terms. a1 0, a2 8, a4 30 a0

In Exercises 6974, find a quadratic model for the sequence with the indicated terms.a0 3, a2 5, a6 57 a1

The table shows the numbers (in thousands) of Alaskan residents from 2005 through 2010. (Source: U.S. Census Bureau) (a) Find the first differences of the data shown in the table. Then find a linear model that approximates the data. Let represent the year, with corresponding to 2005. (b) Use a graphing utility to find a linear model for the data. Compare this model with the model from part (a). (c) Use the models found in parts (a) and (b) to predict the number of residents in 2016. How do these values compare?

HOW DO YOU SEE IT? Find a formula for the sum of the angles (in degrees) of a regular polygon. Then use mathematical induction to prove this formula for a general sided polygon.

In Exercises 77 and 78, determine whether the statement is true or false. Justify your answer. If the statement is true and implies then is also true.

In Exercises 77 and 78, determine whether the statement is true or false. Justify your answer. A sequence with terms has second differences.

The coefficients of a binomial expansion are called ________ ________.

To find binomial coefficients, you can use the ________ ________ or ________ ________.

The symbol used to denote a binomial coefficient is ________ or ________.

When you write the coefficients for a binomial that is raised to a power, you are ________ a ________.

In Exercises 514, find the binomial coefficient. 5C3

In Exercises 514, find the binomial coefficient. 8C6

In Exercises 514, find the binomial coefficient. 12C0

In Exercises 514, find the binomial coefficient. 20C20

In Exercises 514, find the binomial coefficient. 20C15

In Exercises 514, find the binomial coefficient. 12C5

In Exercises 514, find the binomial coefficient. 10 4 

In Exercises 514, find the binomial coefficient.  10 6

In Exercises 514, find the binomial coefficient. 100 98 

In Exercises 514, find the binomial coefficient. 100 2 

In Exercises 1518, evaluate using Pascals Triangle. 6 5

In Exercises 1518, evaluate using Pascals Triangle.  5 2 

In Exercises 1518, evaluate using Pascals Triangle. 7C4

In Exercises 1518, evaluate using Pascals Triangle. 10C2

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression. x 14

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression. x 16 x

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression.a 64

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression. a 55 a

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression. y 43

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression. y 25

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression. x y5 y

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression. c d3 x

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression. 2x y3 c

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression. 7a b3 2

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression. r 3s6

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression. x 2y4 r

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression. 3a 4b5

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression. 2x 5y5

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression. x 2 y24 2

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression. x 2 y 26 x

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression. 1 x y 5

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression.  1 x 2y 6

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression.  2 x y 4

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression.  2 x 3y 5

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression. 2x 34 5x 32 

In Exercises 1940, use the Binomial Theorem to expand and simplify the expression. 4x 13 24x 14 2x

In Exercises 41 44, expand the binomial by using Pascals Triangle to determine the coefficients. 2t s5

In Exercises 41 44, expand the binomial by using Pascals Triangle to determine the coefficients. 3 2z4 2

In Exercises 41 44, expand the binomial by using Pascals Triangle to determine the coefficients. x 2y5 3

In Exercises 41 44, expand the binomial by using Pascals Triangle to determine the coefficients. 3v 26 x

In Exercises 4552, find the specified th term in the expansion of the binomial. x yn 4 ,

In Exercises 4552, find the specified th term in the expansion of the binomial. x yn 7 6

In Exercises 4552, find the specified th term in the expansion of the binomial. x 6yn 3 ,

In Exercises 4552, find the specified th term in the expansion of the binomial. x 10zn 4 7

In Exercises 4552, find the specified th term in the expansion of the binomial. 4x 3yn 8 , 9

In Exercises 4552, find the specified th term in the expansion of the binomial. 5a 6bn 5 5 4x

In Exercises 4552, find the specified th term in the expansion of the binomial. 10x 3yn 10 ,

In Exercises 4552, find the specified th term in the expansion of the binomial. 7x 2yn 7 15 1

In Exercises 5360, find the coefficient of the term in the expansion of the binomial. ax5 x 312a

In Exercises 5360, find the coefficient of the term in the expansion of the binomial. ax8 x 2 312 ax

In Exercises 5360, find the coefficient of the term in the expansion of the binomial. ax 2y8 4x y10 a

In Exercises 5360, find the coefficient of the term in the expansion of the binomial.ax8y 2 x 2y10 a

In Exercises 5360, find the coefficient of the term in the expansion of the binomial. ax4y5 2x 5y9 a

In Exercises 5360, find the coefficient of the term in the expansion of the binomial.ax 6y 2 3x 4y8 a

In Exercises 5360, find the coefficient of the term in the expansion of the binomial. ax8y 6 x 2 y10 ax

In Exercises 5360, find the coefficient of the term in the expansion of the binomial. az4t8 z 2 t10 a

In Exercises 6166, use the Binomial Theorem to expand and simplify the expression. x 5 3 a

In Exercises 6166, use the Binomial Theorem to expand and simplify the expression. 2t 1 3

In Exercises 6166, use the Binomial Theorem to expand and simplify the expression.x 2 3 y1 33

In Exercises 6166, use the Binomial Theorem to expand and simplify the expression. u3 5 25 x

In Exercises 6166, use the Binomial Theorem to expand and simplify the expression. 3t 4 t4

In Exercises 6166, use the Binomial Theorem to expand and simplify the expression. x3 4 2x5 44

In Exercises 6772, simplify the difference quotient, using the Binomial Theorem if necessary. fx hfxh fxx3 f

In Exercises 6772, simplify the difference quotient, using the Binomial Theorem if necessary. fx hfxh fxx4 fx

In Exercises 6772, simplify the difference quotient, using the Binomial Theorem if necessary. fx hfxh fxx6 f

In Exercises 6772, simplify the difference quotient, using the Binomial Theorem if necessary. fx hfxh fxx8 fxx

In Exercises 6772, simplify the difference quotient, using the Binomial Theorem if necessary. fx hfxh fxx f

In Exercises 6772, simplify the difference quotient, using the Binomial Theorem if necessary. fx hfxh fx1 x f

In Exercises 7378, use the Binomial Theorem to expand the complex number. Simplify your result. 1 i4 9

In Exercises 7378, use the Binomial Theorem to expand the complex number. Simplify your result. 2 i5

In Exercises 7378, use the Binomial Theorem to expand the complex number. Simplify your result. 2 3i6

In Exercises 7378, use the Binomial Theorem to expand the complex number. Simplify your result. 5 9 3 2

In Exercises 7378, use the Binomial Theorem to expand the complex number. Simplify your result. 1 2 3 2 i  3

In Exercises 7378, use the Binomial Theorem to expand the complex number. Simplify your result. 5 3i 4

In Exercises 7982, use the Binomial Theorem to approximate the quantity accurate to three decimal places. For example, in Exercise 79, use the expansion 1.028

In Exercises 7982, use the Binomial Theorem to approximate the quantity accurate to three decimal places. For example, in Exercise 79, use the expansion 2.00510

In Exercises 7982, use the Binomial Theorem to approximate the quantity accurate to three decimal places. For example, in Exercise 79, use the expansion 2.9912

In Exercises 7982, use the Binomial Theorem to approximate the quantity accurate to three decimal places. For example, in Exercise 79, use the expansion 1.989

In Exercises 83 and 84, use a graphing utility to graph and in the same viewing window. What is the relationship between the two graphs? Use the Binomial Theorem to write the polynomial function in standard form. fxx gxfx 43 4x, g gf 1

In Exercises 83 and 84, use a graphing utility to graph and in the same viewing window. What is the relationship between the two graphs? Use the Binomial Theorem to write the polynomial function in standard form. fx x gxf x 34 4x 2 1, fxx gxfx

In Exercises 8588, consider independent trials of an experiment in which each trial has two possible outcomes: success or failure. The probability of a success on each trial is and the probability of a failure is In this context, the term in the expansion of gives the probability of successes in the trials of the experiment. You toss a fair coin seven times. To find the probability of obtaining four heads, evaluate the term in the expansion of 2 1 2 7

In Exercises 8588, consider independent trials of an experiment in which each trial has two possible outcomes: success or failure. The probability of a success on each trial is and the probability of a failure is In this context, the term in the expansion of gives the probability of successes in the trials of the experiment. The probability of a baseball player getting a hit during any given time at bat is To find the probability that the player gets three hits during the next 10 times at bat, evaluate the term in the expansion of

In Exercises 8588, consider independent trials of an experiment in which each trial has two possible outcomes: success or failure. The probability of a success on each trial is and the probability of a failure is In this context, the term in the expansion of gives the probability of successes in the trials of the experiment. The probability of a sales representative making a sale with any one customer is The sales representative makes eight contacts a day. To find the probability of making four sales, evaluate the term in the expansion of

In Exercises 8588, consider independent trials of an experiment in which each trial has two possible outcomes: success or failure. The probability of a success on each trial is and the probability of a failure is In this context, the term in the expansion of gives the probability of successes in the trials of the experiment. To find the probability that the sales representative in Exercise 87 makes four sales when the probability of a sale with any one customer is evaluate the term in the expansion of

Finding a Pattern Describe the pattern formed by the sums of the numbers along the diagonal line segments shown in Pascals Triangle (see figure).

Finding a Pattern Use each of the encircled groups of numbers in the figure to form a matrix. Find the determinant of each matrix. Describe the pattern.

Child Support The amounts (in billions of dollars) of child support collected in the United States from 2002 through 2009 can be approximated by the model where represents the year, with corresponding to 2002. (Source: U.S. Department of Health and Human Services) (a) You want to adjust the model so that corresponds to 2007 rather than 2002. To do this, you shift the graph of five units to the left to obtain Use binomial coefficients to write in standard form. (b) Use a graphing utility to graph and in the same viewing window. (c) Use the graphs to estimate when the child support collections exceeded $25 billion.

The table shows the average prices (in cents per kilowatt hour) of residential electricity in the United States from 2003 through 2010. (Source: U.S. Energy Information Administration) (a) Use the regression feature of a graphing utility to find a cubic model for the data. Let represent the year, with corresponding to 2003. (b) Use the graphing utility to plot the data and the model in the same viewing window. (c) You want to adjust the model so that corresponds to 2008 rather than 2003. To do this, you shift the graph of five units to the left to obtain Use binomial coefficients to write in standard form. (d) Use the graphing utility to graph in the same viewing window as (e) Use both models to predict the average price in 2011. Do you obtain the same answer? (f) Do your answers to part (e) seem reasonable? Explain. (g) What factors do you think may have contributed to the change in the average price?

In Exercises 93 and 94, determine whether the statement is true or false. Justify your answer. The Binomial Theorem could be used to produce each row of Pascals Triangle

In Exercises 93 and 94, determine whether the statement is true or false. Justify your answer.A binomial that represents a difference cannot always be accurately expanded using the Binomial Theorem.

Writing In your own words, explain how to form the rows of Pascals Triangle.

Forming Rows of Pascals Triangle Form rows 810 of Pascals Triangle.

Graphical Reasoning Which two functions have identical graphs, and why? Use a graphing utility to graph the functions in the given order and in the same viewing window. Compare the graphs. fx1 x3 98. gx1 x3 fx hx1 3x 3x 2 x3 gx1  kx1 3x 3x 2 x 3 hx1 px1 3x 3x 2 x 3 kx1

HOW DO YOU SEE IT? The expansions of and are as follows. (a) Explain how the exponent of a binomial is related to the number of terms in its expansion. (b) How many terms are in the expansion of x yn? 6xy5 1y6 x y6 1x6 6x5y 15x4y2 20x3y3 15x2y4 5xy4 1y5 x y5 1x5 5x4y 10x3y2 10x2y3 x y4 1x4 4x3y 6x2y2 4xy3 1y4 x y6 x y5 x y, 4, The table

In Exercises 99102, prove the property for all integers and where nr 0 r n. nCr nCn r

In Exercises 99102, prove the property for all integers and where nr 0 r n. nC0 nC1 nC2 . . . nCn 0 n

In Exercises 99102, prove the property for all integers and where nr 0 r n. n1Cr nCr nCr 1 nCn

In Exercises 99102, prove the property for all integers and where nr 0 r n. The sum of the numbers in the th row of Pascals Triangle is 2n

Binomial Coefficients and Pascals Triangle Complete the table and describe the result. What characteristic of Pascals Triangle does this table illustrate?

The ________ ________ ________ states that when there are different ways for one event to occur and different ways for a second event to occur, there are ways for both events to occur.

An ordering of elements is called a ________ of the elements.

The number of permutations of elements taken at a time is given by ________.

The number of ________ ________ of objects is given by n! n1!n2!n3! . . . nk!

When selecting subsets of a larger set in which order is not important, you are finding the number of ________ of elements taken at a time.

The number of combinations of elements taken at a time is given by ________.

In Exercises 714, determine the number of ways a computer can randomly generate one or more such integers from 1 through 12. An odd integer

In Exercises 714, determine the number of ways a computer can randomly generate one or more such integers from 1 through 12. An even integer

In Exercises 714, determine the number of ways a computer can randomly generate one or more such integers from 1 through 12. A prime integer

In Exercises 714, determine the number of ways a computer can randomly generate one or more such integers from 1 through 12. An integer that is greater than 9

In Exercises 714, determine the number of ways a computer can randomly generate one or more such integers from 1 through 12. An integer that is divisible by 4

In Exercises 714, determine the number of ways a computer can randomly generate one or more such integers from 1 through 12. An integer that is divisible by 3

In Exercises 714, determine the number of ways a computer can randomly generate one or more such integers from 1 through 12. wo distinct integers whose sum is 9

In Exercises 714, determine the number of ways a computer can randomly generate one or more such integers from 1 through 12.Two distinct integers whose sum is 8

Entertainment Systems A customer can choose one of three amplifiers, one of two compact disc players, and one of five speaker models for an entertainment system. Determine the number of possible system configurations.

Job Applicants A small college needs two additional faculty members: a chemist and a statistician. There are five applicants for the chemistry position and three applicants for the statistics position. In how many ways can the college fill these positions?

Course Schedule A college student is preparing a course schedule for the next semester. The student may select one of two mathematics courses, one of three science courses, and one of five courses from the social sciences and humanities. How many schedules are possible?

Aircraft Boarding Eight people are boarding an aircraft. Two have tickets for first class and board before those in the economy class. In how many ways can the eight people board the aircraft?

True-False Exam In how many ways can you answer a six-question true-false exam? (Assume that you do not omit any questions.)

True-False Exam In how many ways can you answer a 12-question true-false exam? (Assume that you do not omit any questions.)

License Plate Numbers In the state of Pennsylvania, each standard automobile license plate number consists of three letters followed by a four-digit number. How many distinct license plate numbers can be formed in Pennsylvania?

License Plate Numbers In a certain state, each automobile license plate number consists of two letters followed by a four-digit number. To avoid confusion between O and zero and between I and one, the letters O and I are not used. How many distinct license plate numbers can be formed in this state?

Three-Digit Numbers How many three-digit numbers can you form under each condition? (a) The leading digit cannot be zero. (b) The leading digit cannot be zero and no repetition of digits is allowed. (c) The leading digit cannot be zero and the number must be a multiple of 5. (d) The number is at least 400.

Four-Digit Numbers How many four-digit numbers can you form under each condition? (a) The leading digit cannot be zero. (b) The leading digit cannot be zero and no repetition of digits is allowed. (c) The leading digit cannot be zero and the number must be less than 5000. (d) The leading digit cannot be zero and the number must be even.

Combination Lock A combination lock will open when you select the right choice of three numbers (from 1 to 40, inclusive). How many different lock combinations are possible?

Combination Lock A combination lock will open when you select the right choice of three numbers (from 1 to 50, inclusive). How many different lock combinations are possible?

Concert Seats Four couples have reserved seats in one row for a concert. In how many different ways can they sit when (a) there are no seating restrictions? (b) the two members of each couple wish to sit together?

Single File In how many orders can four girls and four boys walk through a doorway single file when (a) there are no restrictions? (b) the girls walk through before the boys?

Posing for a Photograph In how many ways can five children posing for a photograph line up in a row?

Riding in a Car In how many ways can six people sit in a six-passenger car?

In Exercises 3134, evaluate nPr 4P4

In Exercises 3134, evaluate nPr 8P3

In Exercises 3134, evaluate nPr 20P2

In Exercises 3134, evaluate nPr 5P4

In Exercises 3538, evaluate using a graphing utility. 20P5

In Exercises 3538, evaluate using a graphing utility. 100P5

In Exercises 3538, evaluate using a graphing utility. 100P3

In Exercises 3538, evaluate using a graphing utility. 10P8

A patient with end-stage kidney disease has nine family members who are potential kidney donors. How many possible orders are there for a best match, a second-best match, and a third-best match?

Choosing Officers From a pool of 12 candidates, the offices of president, vice-president, secretary, and treasurer will be filled. In how many different ways can the offices be filled?

Batting Order A baseball coach is creating a nine-player batting order by selecting from a team of 15 players. How many different batting orders are possible?

Athletics Eight sprinters have qualified for the finals in the 100-meter dash at the NCAA national track meet. In how many ways can the sprinters come in first, second, and third? (Assume there are no ties.)

In Exercises 4346, find the number of distinguishable permutations of the group of letters. A, A, G, E, E, E, M

In Exercises 4346, find the number of distinguishable permutations of the group of letters. B, B, B, T, T, T, T, T

In Exercises 4346, find the number of distinguishable permutations of the group of letters. A, L, G, E, B, R, A

In Exercises 4346, find the number of distinguishable permutations of the group of letters. M, I, S, S, I, S, S, I, P, P, I

. Writing Permutations Write all permutations of the letters A, B, C, and D.

Writing Permutations Write all permutations of the letters A, B, C, and D when letters B and C must remain between A and D.

In Exercises 4952, evaluate using the formula from this section. 5C2

In Exercises 4952, evaluate using the formula from this section. 6C3

In Exercises 4952, evaluate using the formula from this section. 6C3

In Exercises 4952, evaluate using the formula from this section. 25C0

In Exercises 5356, evaluate using a graphing utility. 20C4

In Exercises 5356, evaluate using a graphing utility. 10C7

In Exercises 5356, evaluate using a graphing utility. 42C5

In Exercises 5356, evaluate using a graphing utility. 50C6

Writing Combinations Write all combinations of two letters that you can form from the letters A, B, C, D, E, and F. (The order of the two letters is not important.)

Forming an Experimental Group In order to conduct an experiment, researchers randomly select five students from a class of 20. How many different groups of five students are possible?

Jury Selection In how many different ways can a jury of 12 people be randomly selected from a group of 40 people?

Committee Members A U.S. Senate Committee has 14 members. Assuming party affiliation is not a factor in selection, how many different committees are possible from the 100 U.S. senators?

Lottery Choices In the Massachusetts Mass Cash game, a player randomly chooses five distinct numbers from 1 to 35. In how many ways can a player select the five numbers?

Lottery Choices In the Louisiana Lotto game, a player randomly chooses six distinct numbers from 1 to 40. In how many ways can a player select the six numbers?

Defective Units A shipment of 25 television sets contains three defective units. In how many ways can a vending company purchase four of these units and receive (a) all good units, (b) two good units, and (c) at least two good units?

Interpersonal Relationships The complexity of interpersonal relationships increases dramatically as the size of a group increases. Determine the numbers of different two-person relationships in groups of people of sizes (a) 3, (b) 8, (c) 12, and (d) 20.

Poker Hand You are dealt five cards from a standard deck of 52 playing cards. In how many ways can you get (a) a full house and (b) a five-card combination containing two jacks and three aces? (A full house consists of three of one kind and two of another. For example, A-A-A-5-5 and K-K-K-10-10 are full houses.)

Job Applicants An employer interviews 12 people for four openings at a company. Five of the 12 people are women. All 12 applicants are qualified. In how many ways can the employer fill the four positions when (a) the selection is random and (b) exactly two selections are women?

Forming a Committee A local college is forming a six-member research committee having one administrator, three faculty members, and two students. There are seven administrators, 12 faculty members, and 20 students in contention for the committee. How many six-member committees are possible?

Law Enforcement A police department uses computer imaging to create digital photographs of alleged perpetrators from eyewitness accounts. One software package contains 195 hairlines, 99 sets of eyes and eyebrows, 89 noses, 105 mouths, and 74 chins and cheek structures. (a) Find the possible number of different faces that the software could create. (b) An eyewitness can clearly recall the hairline and eyes and eyebrows of a suspect. How many different faces can be produced with this information?

In Exercises 6972, find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of the polygon.) Pentagon

In Exercises 6972, find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of the polygon.) Hexagon

In Exercises 6972, find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of the polygon.) Octagon.

In Exercises 6972, find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of the polygon.) Decagon (10 sides

Geometry Three points that are not collinear determine three lines. How many lines are determined by nine points, no three of which are collinear?

Lottery Powerball is a lottery game that is operated by the Multi-State Lottery Association and is played in 42 states, Washington D.C., and the U.S. Virgin Islands. The game is played by drawing five white balls out of a drum of 59 white balls (numbered 159) and one red powerball out of a drum of 35 red balls (numbered 135). The jackpot is won by matching all five white balls in any order and the red powerball. (a) Find the possible number of winning Powerball numbers. (b) Find the possible number of winning Powerball numbers when you win the jackpot by matching all five white balls in order and the red powerball.

In Exercises 7582, solve for n 4 n1P2 n2P3 5

In Exercises 7582, solve for n 5 n1P1 nP2

In Exercises 7582, solve for n n1P3 n2P3 6 n2P1 4 nP2 4

In Exercises 7582, solve for n n2P3 6 n2P1 4

In Exercises 7582, solve for n 4 nP3 n2P4

In Exercises 7582, solve for n nP5 18 n2P4 1

In Exercises 7582, solve for n nP4 nP6 12 n1P5 10 n1P3 nP

In Exercises 7582, solve for n nP6 12 n1P5 1

In Exercises 83 and 84, determine whether the statement is true or false. Justify your answer. The number of letter pairs that can be formed in any order from any two of the first 13 letters in the alphabet (AM) is an example of a permutation

In Exercises 83 and 84, determine whether the statement is true or false. Justify your answer.The number of permutations of elements can be determined by using the Fundamental Counting Principle.

Think About It Without calculating the numbers, determine which of the following is greater. Explain. (a) The number of combinations of 10 elements taken six at a time (b) The number of permutations of 10 elements taken six at a time

HOW DO YOU SEE IT? Without calculating, determine whether the value of is greater than the value of for the values of and given in the table. Complete the table using yes (Y) or no (N). Is the value of always greater than the value of Explain.

n Exercises 8790, prove the identity. nPn 1 nPn

n Exercises 8790, prove the identity. Cn nC0

n Exercises 8790, prove the identity. nCn 1 nC1

n Exercises 8790, prove the identity. nCr nPr

Think About It Can your graphing utility evaluate If not, then explain why.

An ________ is any happening for which the result is uncertain, and the possible results are called ________.

The set of all possible outcomes of an experiment is called the ________ ________.

To determine the ________ of an event, you can use the formula where is the number of outcomes in the event and is the number of outcomes in the sample space.

If then is an ________ event, and if then is a ________ event.

If two events from the same sample space have no outcomes in common, then the two events are ________ ________.

If the occurrence of one event has no effect on the occurrence of a second event, then the events are ________.

The ________ of an event is the collection of all outcomes in the sample space that are not in A

Match the probability formula with the correct probability name. (a) Probability of the union of two events (i) (b) Probability of mutually exclusive events (ii) (c) Probability of independent events (iii) (d) Probability of a complement PA BPAPB A PA1 PA PA BPAPBPA BP PA and BPAPB

In Exercises 914, find the sample space for the experiment. You toss a coin and a six-sided die

In Exercises 914, find the sample space for the experiment. You toss a six-sided die twice and record the sum of the results

In Exercises 914, find the sample space for the experiment. A taste tester ranks three varieties of yogurt, A, B, and C, according to preference.

In Exercises 914, find the sample space for the experiment. You select two marbles (without replacement) from a bag containing two red marbles, two blue marbles, and one yellow marble. You record the color of each marble.

In Exercises 914, find the sample space for the experiment. Two county supervisors are selected from five supervisors, A, B, C, D, and E, to study a recycling plan.

In Exercises 914, find the sample space for the experiment. A sales representative makes presentations about a product in three homes per day. In each home, there may be a sale (denote by S) or there may be no sale (denote by F).

In Exercises 1520, find the probability for the experiment of tossing a coin three times. Use the sample space S {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. The probability of getting exactly one tail

In Exercises 1520, find the probability for the experiment of tossing a coin three times. Use the sample space S {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. The probability of getting exactly two tails

In Exercises 1520, find the probability for the experiment of tossing a coin three times. Use the sample space S {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. The probability of getting a head on the first toss

In Exercises 1520, find the probability for the experiment of tossing a coin three times. Use the sample space S {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. The probability of getting a tail on the last toss

In Exercises 1520, find the probability for the experiment of tossing a coin three times. Use the sample space S {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. The probability of getting at least one head

In Exercises 1520, find the probability for the experiment of tossing a coin three times. Use the sample space S {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. The probability of getting at least two heads

In Exercises 2124, find the probability for the experiment of drawing a card at random from a standard deck of 52 playing cards. The card is a face card.

In Exercises 2124, find the probability for the experiment of drawing a card at random from a standard deck of 52 playing cards. The card is not a face card.

In Exercises 2124, find the probability for the experiment of drawing a card at random from a standard deck of 52 playing cards. The card is a red face card.

In Exercises 2124, find the probability for the experiment of drawing a card at random from a standard deck of 52 playing cards. The card is a 9 or lower. (Aces are low.)

In Exercises 2530, find the probability for the experiment of tossing a six-sided die twice. The sum is 6

In Exercises 2530, find the probability for the experiment of tossing a six-sided die twice. The sum is at least 8.

In Exercises 2530, find the probability for the experiment of tossing a six-sided die twice. The sum is less than 11.

In Exercises 2530, find the probability for the experiment of tossing a six-sided die twice. The sum is 2, 3, or 12.

In Exercises 2530, find the probability for the experiment of tossing a six-sided die twice. The sum is odd and no more than 7.

In Exercises 2530, find the probability for the experiment of tossing a six-sided die twice. The sum is odd or prime.

In Exercises 3134, find the probability for the experiment of drawing two marbles at random (without replacement) from a bag containing one green, two yellow, and three red marbles. Both marbles are red.

In Exercises 3134, find the probability for the experiment of drawing two marbles at random (without replacement) from a bag containing one green, two yellow, and three red marbles. Both marbles are yellow.

In Exercises 3134, find the probability for the experiment of drawing two marbles at random (without replacement) from a bag containing one green, two yellow, and three red marbles. Neither marble is yellow.

In Exercises 3134, find the probability for the experiment of drawing two marbles at random (without replacement) from a bag containing one green, two yellow, and three red marbles. The marbles are different colors.

Graphical Reasoning In 2011, there were approximately 13.75 million unemployed workers in the United States. The circle graph shows the age profile of these unemployed workers. (Source: U.S. Bureau of Labor Statistics) (a) Estimate the number of unemployed workers in the 1619 age group. (b) What is the probability that a person selected at random from the population of unemployed workers is in the 2544 age group? (c) What is the probability that a person selected at random from the population of unemployed workers is in the 4564 age group? (d) What is the probability that a person selected at random from the population of unemployed workers is 45 or older?

Data Analysis An independent polling organization interviews one hundred college students to determine their political party affiliations and whether they favor a balanced-budget amendment to the Constitution. The table lists the results of the study. In the table, represents Democrat and represents Republican. Find the probability that a person selected at random from the sample is as described. (a) A person who does not favor the amendment (b) A Republican (c) A Democrat who favors the amendment

Education In a high school graduating class of 128 students, 52 are on the honor roll. Of these, 48 are going on to college; of the other 76 students, 56 are going on to college. What is the probability that a student selected at random from the class is (a) going to college, (b) not going to college, and (c) not going to college and on the honor roll?

Alumni Association A college sends a survey to members of the class of 2012. Of the 1254 people who graduated that year, 672 are women, of whom 124 went on to graduate school. Of the 582 male graduates, 198 went on to graduate school. What is the probability that a class of 2012 alumnus selected at random is (a) female, (b) male, and (c) female and did not attend graduate school?

Winning an Election Three people are running for president of a class. The results of a poll indicate that the first candidate has an estimated 37% chance of winning and the second candidate has an estimated 44% chance of winning. What is the probability that the third candidate will win?

Payroll Error The employees of a company work in six departments: 31 are in sales, 54 are in research, 42 are in marketing, 20 are in engineering, 47 are in finance, and 58 are in production. The payroll department loses one employees paycheck. What is the probability that the employee works in the research department?

In Exercises 4148, the sample spaces are large and you should use the counting principles discussed in Section 8.6. A class receives a list of 20 study problems, from which 10 will be part of an upcoming exam. A student knows how to solve 15 of the problems. Find the probability that the student will be able to answer (a) all 10 questions on the exam, (b) exactly eight questions on the exam, and (c) at least nine questions on the exam.

In Exercises 4148, the sample spaces are large and you should use the counting principles discussed in Section 8.6. A payroll department addresses five paychecks and envelopes to five different people and randomly inserts the paychecks into the envelopes. What is the probability that (a) exactly one paycheck is inserted in the correct envelope and (b) at least one paycheck is in the correct envelope?

In Exercises 4148, the sample spaces are large and you should use the counting principles discussed in Section 8.6. On a game show, you are given five digits to arrange in the proper order to form the price of a car. If you are correct, then you win the car. What is the probability of winning, given the following conditions? (a) You guess the position of each digit. (b) You know the first digit and guess the positions of the other digits.

In Exercises 4148, the sample spaces are large and you should use the counting principles discussed in Section 8.6. The deck for a card game is made up of 108 cards. Twenty-five each are red, yellow, blue, and green, and eight are wild cards. Each player is randomly dealt a seven-card hand. (a) What is the probability that a hand will contain exactly two wild cards? (b) What is the probability that a hand will contain two wild cards, two red cards, and three blue cards?

In Exercises 4148, the sample spaces are large and you should use the counting principles discussed in Section 8.6. You draw one card at random from a standard deck of 52 playing cards. Find the probability that (a) the card is an even-numbered card, (b) the card is a heart or a diamond, and (c) the card is a nine or a face card.

In Exercises 4148, the sample spaces are large and you should use the counting principles discussed in Section 8.6. You draw five cards at random from a standard deck of 52 playing cards. What is the probability that the hand drawn is a full house? (A full house is a hand that consists of two of one kind and three of another kind.)

In Exercises 4148, the sample spaces are large and you should use the counting principles discussed in Section 8.6. A shipment of 12 microwave ovens contains three defective units. A vending company has ordered four units, and because each has identical packaging, the selection will be random. What is the probability that (a) all four units are good, (b) exactly two units are good, and (c) at least two units are good?

In Exercises 4148, the sample spaces are large and you should use the counting principles discussed in Section 8.6. 48. ATM personal identification number (PIN) codes typically consist of four-digit sequences of numbers. Find the probability that if you forget your PIN, then you can guess the correct sequence (a) at random and (b) when you recall the first two digits.

Random Number Generator A random number generator on a computer selects two integers from 1 through 40. What is the probability that (a) both numbers are even, (b) one number is even and one number is odd, (c) both numbers are less than 30, and (d) the same number is selected twice?

Flexible Work Hours In a survey, people were asked whether they would prefer to work flexible hourseven when it meant slower career advancementso they could spend more time with their families. The figure shows the results of the survey. What is the probability that three people chosen at random would prefer flexible work hours?

In Exercises 5154, you are given the probability that an event will happen. Find the probability that the event will not happen. PE0.87

In Exercises 5154, you are given the probability that an event will happen. Find the probability that the event will not happen. PE0.36

In Exercises 5154, you are given the probability that an event will happen. Find the probability that the event will not happen. P 3 E1 4

In Exercises 5154, you are given the probability that an event will happen. Find the probability that the event will not happen. PE2 P 3

In Exercises 5558, you are given the probability that an event will not happen. Find the probability that the event will happen. PE0.23

In Exercises 5558, you are given the probability that an event will not happen. Find the probability that the event will happen. PE0.92

In Exercises 5558, you are given the probability that an event will not happen. Find the probability that the event will happen. P 100 E17 35

In Exercises 5558, you are given the probability that an event will not happen. Find the probability that the event will happen. PE61 P 100

A space vehicle has an independent backup system for one of its communication networks. The probability that either system will function satisfactorily during a flight is 0.985. What is the probability that during a given flight (a) both systems function satisfactorily, (b) at least one system functions satisfactorily, and (c) both systems fail?

Backup Vehicle A fire company keeps two rescue vehicles. Because of the demand on the vehicles and the chance of mechanical failure, the probability that a specific vehicle is available when needed is 90%. The availability of one vehicle is independent of the availability of the other. Find the probability that (a) both vehicles are available at a given time, (b) neither vehicle is available at a given time, and (c) at least one vehicle is available at a given time.

Roulette American roulette is a game in which a wheel turns on a spindle and is divided into 38 pockets. Thirty-six of the pockets are numbered 136, of which half are red and half are black. Two of the pockets are green and are numbered 0 and 00 (see figure). The dealer spins the wheel and a small ball in opposite directions. As the ball slows to a stop, it has an equal probability of landing in any of the numbered pockets. (a) Find the probability of landing in the number 00 pocket. (b) Find the probability of landing in a red pocket. (c) Find the probability of landing in a green pocket or a black pocket. (d) Find the probability of landing in the number 14 pocket on two consecutive spins. (e) Find the probability of landing in a red pocket on three consecutive spins.

Boy or a Girl? Assume that the probability of the birth of a child of a particular sex is 50%. In a family with four children, what is the probability that (a) all the children are boys, (b) all the children are the same sex, and (c) there is at least one boy?

Geometry You and a friend agree to meet at your favorite fast-food restaurant between 5:00 P.M. and 6:00 P.M. The one who arrives first will wait 15 minutes for the other, and then will leave (see figure). What is the probability that the two of you will actually meet, assuming that your arrival times are random within the

Estimating You drop a coin of diameter onto a paper that contains a grid of squares units on a side (see figure). (a) Find the probability that the coin covers a vertex of one of the squares on the grid. (b) Perform the experiment 100 times and use the results to approximate

In Exercises 65 and 66, determine whether the statement is true or false. Justify your answer. If and are independent events with nonzero probabilities, then can occur when occurs.

In Exercises 65 and 66, determine whether the statement is true or false. Justify your answer.Rolling a number less than 3 on a normal six-sided die has a probability of . The complement of this event is to roll a number greater than 3, and its probability is

Pattern Recognition Consider a group of people. (a) Explain why the following pattern gives the probabilities that the people have distinct birthdays. (b) Use the pattern in part (a) to write an expression for the probability that people have distinct birthdays. (c) Let be the probability that the people have distinct birthdays. Verify that this probability can be obtained recursively by and (d) Explain why gives the probability that at least two people in a group of people have the same birthday. (e) Use the results of parts (c) and (d) to complete the table. (f) How many people must be in a group so that the probability of at least two of them having the same birthday is greater than Explain.

HOW DO YOU SEE IT? The circle graphs show the percents of undergraduate students by class level at two colleges. A student is chosen at random from the combined undergraduate population of the two colleges. The probability that the student is a freshman, sophomore, or junior is 81%. Which college has a greater number of undergraduate students? Explain your reasoning.

In Exercises 14, write the first five terms of the sequence. (Assume that begins with 1.) an 2  6 n

In Exercises 14, write the first five terms of the sequence. (Assume that begins with 1.) an 1n5n 2n 1 an 2

In Exercises 14, write the first five terms of the sequence. (Assume that begins with 1.) a an nn 1n 72 n! an

In Exercises 14, write the first five terms of the sequence. (Assume that begins with 1.) an nn 1n 72

In Exercises 58, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) 2, 2, 2, 2, 2, .

In Exercises 58, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) 1, 2, 7, 14, 23, .

In Exercises 58, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) 4, 2, , . . . 4 3, 1, 4

In Exercises 58, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) 1, 1 2, 1 3, 1 4, 1 5 4, 2, , .

In Exercises 912, simplify the factorial expression. 9!

In Exercises 912, simplify the factorial expression. 4! 0!

In Exercises 912, simplify the factorial expression. 3! 5! 6!

In Exercises 912, simplify the factorial expression. 7! 6! 6! 8!

In Exercises 13 and 14, find the sum.  4 j 1 6 j 2

In Exercises 13 and 14, find the sum.  10 k1 2k3

In Exercises 15 and 16, use sigma notation to write the sum. 1 21  1 22  1 23 . . .  1 220  10

In Exercises 15 and 16, use sigma notation to write the sum. 1 2  2 3  3 4 . . .  9 10

In Exercises 17 and 18, find the sum of the infinite series.   i1 4 10i

In Exercises 17 and 18, find the sum of the infinite series.   k1 2 1 100 k

Compound Interest An investor deposits $10,000 in an account that earns 2.25% interest compounded monthly. The balance in the account after months is given by (a) Write the first 10 terms of the sequence. (b) Find the balance in the account after 10 years by computing the 120th term of the sequence

Lottery Ticket Sales The total sales (in billions of dollars) of lottery tickets in the United States from 2001 through 2010 can be approximated by the model where is the year, with corresponding to 2001. Write the terms of this finite sequence. Use a graphing utility to construct a bar graph that represents the sequence. (Source: TLF Publications, Inc.)

In Exercises 2124, determine whether the sequence is arithmetic. If so, then find the common difference. 6, 1, 8, 15, 22, . . . 8.2

In Exercises 2124, determine whether the sequence is arithmetic. If so, then find the common difference. 0, 1, 3, 6, 10, .

In Exercises 2124, determine whether the sequence is arithmetic. If so, then find the common difference. 1, 2

In Exercises 2124, determine whether the sequence is arithmetic. If so, then find the common difference. 1, 15 16, 7 8, 13 16, 3 4,

In Exercises 2528, find a formula for for the arithmetic sequence a1 7, d 12 a1

In Exercises 2528, find a formula for for the arithmetic sequence a1 28, d 5a

In Exercises 2528, find a formula for for the arithmetic sequence a2 93, a6 65 a

In Exercises 2528, find a formula for for the arithmetic sequence a7 8, a13 6

In Exercises 29 and 30, write the first five terms of the arithmetic sequence. a1 3, d 11 a

In Exercises 29 and 30, write the first five terms of the arithmetic sequence. a1 25, ak1 ak 3

Sum of a Finite Arithmetic Sequence Find the sum of the first 100 positive multiples of 7.

Sum of a Finite Arithmetic Sequence Find the sum of the integers from 40 to 90 (inclusive).

In Exercises 3336, find the partial sum. 20 3j 10 j1 2j 3 a1 3,

In Exercises 3336, find the partial sum.  8 j1 20 3

In Exercises 3336, find the partial sum. 11 k1  2 3 k 4

In Exercises 3336, find the partial sum.  25 k1  3k 1 4

Job Offer The starting salary for a job is $43,800 with a guaranteed increase of $1950 per year. Determine (a) the salary during the fifth year and (b) the total compensation through five full years of employment.

Baling Hay In the first two trips baling hay around a large field, a farmer obtains 123 bales and 112 bales, respectively. Because each round gets shorter, the farmer estimates that the same pattern will continue. Estimate the total number of bales made after the farmer takes another six trips around the field.

In Exercises 3942, determine whether the sequence is geometric. If so, then find the common ratio. 6, 12, 24, 48,

In Exercises 3942, determine whether the sequence is geometric. If so, then find the common ratio. 54, 18, 6, 2, . .

In Exercises 3942, determine whether the sequence is geometric. If so, then find the common ratio. 1 5, 3 5, 9 5, 27 5 ,

In Exercises 3942, determine whether the sequence is geometric. If so, then find the common ratio. 1 4, 2 5, 3 6, 4 7, .

In Exercises 4346, write the first five terms of the geometric sequence. a1 2, 4 r 15 1

In Exercises 4346, write the first five terms of the geometric sequence. a1 4, r 1 a1 2, 4 r 1

In Exercises 4346, write the first five terms of the geometric sequence. a1 9, a3 4 a

In Exercises 4346, write the first five terms of the geometric sequence. a1 2, a3 12

In Exercises 4750, write an expression for the th term of the geometric sequence. Then find the 10th term of the sequence. a1 18, a2 9 a3

In Exercises 4750, write an expression for the th term of the geometric sequence. Then find the 10th term of the sequence. a3 6, a4 1

In Exercises 4750, write an expression for the th term of the geometric sequence. Then find the 10th term of the sequence. a1 100, r 1.05 a

In Exercises 4750, write an expression for the th term of the geometric sequence. Then find the 10th term of the sequence. a1 5, r 0.2

In Exercises 5156, find the sum of the finite geometric sequence.  7 i1 2i1

In Exercises 5156, find the sum of the finite geometric sequence.  5 i1 3i1

In Exercises 5156, find the sum of the finite geometric sequence.  4 i1 1 2 i

In Exercises 5156, find the sum of the finite geometric sequence.  6 i1 1 3 i1 

In Exercises 5156, find the sum of the finite geometric sequence.  5 i1 2i1

In Exercises 5156, find the sum of the finite geometric sequence.  4 i1 63i 

In Exercises 57 and 58, use a graphing utility to find the sum of the finite geometric sequence.  10 i1 10 3 5 i1 

In Exercises 57 and 58, use a graphing utility to find the sum of the finite geometric sequence.  15 i1 200.2i1 

In Exercises 5962, find the sum of the infinite geometric series.   i0 7 8 i

In Exercises 5962, find the sum of the infinite geometric series.   i0 0.5i 

In Exercises 5962, find the sum of the infinite geometric series.   k1 4 2 3 k1 

In Exercises 5962, find the sum of the infinite geometric series.   k1 1.31 10 k1 

Depreciation A paper manufacturer buys a machine for $120,000. During the next 5 years, it will depreciate at a rate of 30% per year. (In other words, at the end of each year the depreciated value will be 70% of what it was at the beginning of the year.) (a) Find the formula for the th term of a geometric sequence that gives the value of the machine full years after it was purchased. (b) Find the depreciated value of the machine after 5 full years.

Annuity An investor deposits $800 in an account on the first day of each month for 10 years. The account pays 3%, compounded monthly. What will the balance be at the end of 10 years?

In Exercises 6568, use mathematical induction to prove the formula for every positive integer n 3 5 7 . . . 2n 1nn 2 n.

In Exercises 6568, use mathematical induction to prove the formula for every positive integer n 1  3 2 2  5 2 . . .  1 2 n 1n 4 n 3 3 

In Exercises 6568, use mathematical induction to prove the formula for every positive integer n  n1 i0 ari a1 rn 1 r 1 

In Exercises 6568, use mathematical induction to prove the formula for every positive integer n  n1 k0 a kdn 2 2a n 1d  n1

In Exercises 6972, use mathematical induction to find a formula for the sum of the first terms of the sequence. 9, 13, 17, 21, . .

In Exercises 6972, use mathematical induction to find a formula for the sum of the first terms of the sequence. 68, 60, 52, 44

In Exercises 6972, use mathematical induction to find a formula for the sum of the first terms of the sequence. 1

In Exercises 6972, use mathematical induction to find a formula for the sum of the first terms of the sequence. 12 1, ,

In Exercises 73 and 74, find the sum using the formulas for the sums of powers of integers.  75 n1 n

In Exercises 73 and 74, find the sum using the formulas for the sums of powers of integers.  6 n1 n5 n2  75

In Exercises 75 and 76, write the first five terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a perfect linear model, a perfect quadratic model, or neither. a1 5 an an1 5 an

In Exercises 75 and 76, write the first five terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a perfect linear model, a perfect quadratic model, or neither. 75. 76 a1 3 an an1 2n a

In Exercises 77 and 78, find the binomial coefficient. 6C4

In Exercises 77 and 78, find the binomial coefficient. 12C3

In Exercises 79 and 80, evaluate using Pascals Triangle. 7 2

In Exercises 79 and 80, evaluate using Pascals Triangle. 10 4

In Exercises 8184, use the Binomial Theorem to expand and simplify the expression. Remember that i  1. x 44

In Exercises 8184, use the Binomial Theorem to expand and simplify the expression. Remember that i  1. a 3b5 x

In Exercises 8184, use the Binomial Theorem to expand and simplify the expression. Remember that i  1. 5 2i4

In Exercises 8184, use the Binomial Theorem to expand and simplify the expression. Remember that i  1. 4 5i3 5

Numbers in a Hat You place slips of paper numbered 1 through 14 in a hat. In how many ways can you draw two numbers at random with replacement that total 12?

Shopping A customer in an electronics store can choose one of six speaker systems, one of five DVD players, and one of six flat screen televisions to design a home theater system. How many systems can the customer design?

Telephone Numbers All of the land line telephone numbers in a small town use the same three-digit prefix. How many different telephone numbers are possible by changing only the last four digits?

Course Schedule A college student is preparing a course schedule for the next semester. The student may select one of three mathematics courses, one of four science courses, and one of six history courses. How many schedules are possible?

Genetics A geneticist is using gel electrophoresis to analyze five DNA samples. The geneticist treats each sample with a different restriction enzyme and then injects it into one of five wells formed in a bed of gel. In how many orders can the geneticist inject the five samples into the wells?

Race There are 10 bicyclists entered in a race. In how many different ways could the top 3 places be decided?

Jury Selection A group of potential jurors has been narrowed down to 32 people. In how many ways can a jury of 12 people be selected?

Menu Choices A local sub shop offers five different breads, four different meats, three different cheeses, and six different vegetables. You can choose one bread and any number of the other items. Find the total number of combinations of sandwiches possible.

Apparel A man has five pairs of socks, of which no two pairs are the same color. He randomly selects two socks from a drawer. What is the probability that he gets a matched pair?

Bookshelf Order A child returns a five-volume set of books to a bookshelf. The child is not able to read, and so cannot distinguish one volume from another. What is the probability that the child shelves the books in the correct order?

Students by Class At a university, 31% of the students are freshmen, 26% are sophomores, 25% are juniors, and 18% are seniors. One student receives a cash scholarship randomly by lottery. What is the probability that the scholarship winner is (a) a junior or senior? (b) a freshman, sophomore, or junior?

Data Analysis Interviewers asked a sample of college students, faculty members, and administrators whether they favored a proposed increase in the annual activity fee to enhance student life on campus. The table lists the results. Find the probability that a person selected at random from the sample is as described. (a) Not in favor of the proposal (b) A student (c) A faculty member in favor of the proposal

Tossing a Die You toss a six-sided die four times. What is the probability of getting a 5 on each roll?

Tossing a Die You toss a six-sided die six times. What is the probability of getting each number exactly once?

Drawing a Card You randomly draw a card from a standard deck of 52 playing cards. What is the probability that the card is not a club?

Tossing a Coin Find the probability of obtaining at least one tail when you toss a coin five times.

In Exercises 101104, determine whether the statement is true or false. Justify your answer. n 2! n! n 2 n

In Exercises 101104, determine whether the statement is true or false. Justify your answer.  5 i1 i 3 2i 5 i1 i3  5 i1 2i n 2

In Exercises 101104, determine whether the statement is true or false. Justify your answer.  8 k1 3k 3 8 k1 k 

In Exercises 101104, determine whether the statement is true or false. Justify your answer.  6 j1 2j  8 j3 2j2 

Think About It An infinite sequence is a function. What is the domain of the function?

Think About It How do the two sequences differ? an 1n n an 1n1 n an

Writing Explain what is meant by a recursion formula.

Writing Write a brief paragraph explaining how to identify the graph of an arithmetic sequence and the graph of a geometric sequence.