Solution: For 61-66, use a graphing utility to find the sum

For the function f(x) = x - I , find f(2) and f(3). (pp. 208-219)

True or False A function is a relation between two sets D and R so that each element x in the first set D is related to exactly one element y in the second set R. (pp. 208-219)

A(n) __ is a function whose domain is the set of positive integers.

For the sequence {sn} = {4n - I}, the first term is Sl = ___ and the fourth term is S4 = __ .

L (2k) =___. k=l

True or False Sequences are sometimes defined recursively.

True or False A sequence is a function.

True or False L k = 3 k=1

In Problems 9-14, evaluate each factorial expression. 10!

In Problems 9-14, evaluate each factorial expression. 9!

In Problems 9-14, evaluate each factorial expression. 9! 6!

In Problems 9-14, evaluate each factorial expression. 12! 1O!

In Problems 9-14, evaluate each factorial expression. 3! 7! -4!

In Problems 9-14, evaluate each factorial expression. 5! 8! -3!

In Problems 15-26, write down the first five terms of each sequence. (snl = {n}

In Problems 15-26, write down the first five terms of each sequence. (sill = {n2 + I}

In Problems 15-26, write down the first five terms of each sequence. (alll = {_n } n+2

In Problems 15-26, write down the first five terms of each sequence. (bill = --2n

In Problems 15-26, write down the first five terms of each sequence. (cnl = {(-1)"- 1 n2

In Problems 15-26, write down the first five terms of each sequence. (dill = {(-1)"- 1( 2n n _ I )}

In Problems 15-26, write down the first five terms of each sequence. (sill = 3 n + 1

In Problems 15-26, write down the first five terms of each sequence. sill = {I}

In Problems 15-26, write down the first five terms of each sequence. (till = (n + l)(n + 2)

In Problems 15-26, write down the first five terms of each sequence. (alll = -;; 1l }

In Problems 15-26, write down the first five terms of each sequence. (bill = {;,}

In Problems 15-26, write down the first five terms of each sequence. (clll = {I}

In Problems 27-34, the given pattern continues. Write down the nth term of a sequence (anl suggested by the pattern. 1 2 3 4 2' 3< 4'< 5

In Problems 27-34, the given pattern continues. Write down the nth term of a sequence (anl suggested by the pattern. 1 1 1 1 1 .2 ' 2 . 3 ' 3 .4 ' 4

In Problems 27-34, the given pattern continues. Write down the nth term of a sequence (anl suggested by the pattern. 1 1 1 2 4 8

In Problems 27-34, the given pattern continues. Write down the nth term of a sequence (anl suggested by the pattern. 2 4 8 16 3' 9' 27 ' 81 "

In Problems 27-34, the given pattern continues. Write down the nth term of a sequence (anl suggested by the pattern. 1, -1, 1, -1, 1, -1, ...

In Problems 27-34, the given pattern continues. Write down the nth term of a sequence (anl suggested by the pattern. 1 1 1 1 1 ' 2, 3' 4, 5 ' 6 , 7' 8" "

In Problems 27-34, the given pattern continues. Write down the nth term of a sequence (anl suggested by the pattern. 1, -2, 3, -4, 5, -6, ...

In Problems 27-34, the given pattern continues. Write down the nth term of a sequence (anl suggested by the pattern. 2, -4, 6, -8, 10, ...

In Problems 35-48, a sequence is defined recursively. Write down the first five terms. al = 2; an = 3 + an-I

In Problems 35-48, a sequence is defined recursively. Write down the first five terms. al = 3; all = 4 - all-l

In Problems 35-48, a sequence is defined recursively. Write down the first five terms. a] = -2; an = n + all-l

In Problems 35-48, a sequence is defined recursively. Write down the first five terms. al = 1; an = n - an-I

In Problems 35-48, a sequence is defined recursively. Write down the first five terms. al = 5; all = 2all-1

In Problems 35-48, a sequence is defined recursively. Write down the first five terms. al = 2; all = -an-l

In Problems 35-48, a sequence is defined recursively. Write down the first five terms. al = 3; an-I an = -n

In Problems 35-48, a sequence is defined recursively. Write down the first five terms. al = -2; all = n + 3all-l

In Problems 35-48, a sequence is defined recursively. Write down the first five terms. al = 1; a1 = ; an = an-1 an-2

In Problems 35-48, a sequence is defined recursively. Write down the first five terms. a1 = -1; a2 = 1; an = an-2 + nan-1

In Problems 35-48, a sequence is defined recursively. Write down the first five terms. al = A ; all = all-l + d

In Problems 35-48, a sequence is defined recursively. Write down the first five terms. a] = A; all = ran-I , r '1= 0

In Problems 35-48, a sequence is defined recursively. Write down the first five terms. a1 = \12; all = V2 + an-l

In Problems 35-48, a sequence is defined recursively. Write down the first five terms. al = v2; all - 2

In Problems 49-58, write out each sum. 11 n L (k + 2) k=l 11

In Problems 49-58, write out each sum. L (2k + 1)

In Problems 49-58, write out each sum. L- k2 k = 11

In Problems 49-58, write out each sum. n 1 L (k + 1)2

In Problems 49-58, write out each sum. n 1 k = 0 3k

In Problems 49-58, write out each sum. 11 CY n-1 1 L - k=O

In Problems 49-58, write out each sum. n-1 1 L k+l k=o 3

In Problems 49-58, write out each sum. 11-1 L (2k + 1) k=O

In Problems 49-58, write out each sum. 11 n k ln k k=2

In Problems 49-58, write out each sum. L (_1)k+12k k=3

In Problems 59-68, express each sum using summation notation. 1 + 2 + 3 + . . . + 20

In Problems 59-68, express each sum using summation notation. 13 + 23 + 33 + ... + 83

In Problems 59-68, express each sum using summation notation. 1 2 3 13 2 + 3 + 4 + .. . + 13 + 1

In Problems 59-68, express each sum using summation notation. 1 + 3 + 5 + 7 + . .. + [2(12) - 1J

In Problems 59-68, express each sum using summation notation. - - + - --+ . . . + (-1) - 3 9 27 36

In Problems 59-68, express each sum using summation notation. 2 4 8 ( 2 ) 11 3 - '9 + 27 - . .. + (-1 ) 12 3

In Problems 59-68, express each sum using summation notation. 3 + - + - + ... + - 2 3 n

In Problems 59-68, express each sum using summation notation. 1 2 3 n - + - + - + ... + - e e2 e3 e

In Problems 59-68, express each sum using summation notation. a + (a + d) + (a + 2d) + ... + (a + nd)

In Problems 59-68, express each sum using summation notation. a + ar + ar2 + ... + arn-J

In Problems 69-80, find the sum of each sequence. 2: 5 k=l

In Problems 69-80, find the sum of each sequence. 50 8 k=l

In Problems 69-80, find the sum of each sequence. 40 2:k k = l

In Problems 69-80, find the sum of each sequence. 24 2: (-k) k= l

In Problems 69-80, find the sum of each sequence. 20 2 6 2: (3k - 7) k=l

In Problems 69-80, find the sum of each sequence. 2 6 2: (3k - 7) k=1

In Problems 69-80, find the sum of each sequence. 2: (k2 + 4) k=1

In Problems 69-80, find the sum of each sequence. 14 2: (k2 - 4) k=O

In Problems 69-80, find the sum of each sequence. 2: (2k) k=lO

In Problems 69-80, find the sum of each sequence. 40 2: (-3k) k=8

In Problems 69-80, find the sum of each sequence. 20 2: k3 k=5

In Problems 69-80, find the sum of each sequence. 24 2: k3 k=4

Credit Card Debt John has a balance of $3000 on his Discover card that charges 1 % interest per month on any unpaid balance. John can afford to pay $100 toward the balance each month. His balance each month after making a $100 payment is given by the recursively defined sequence Bo = $3000, Bn = 1.01Bn-J - 100 Determine John's balance after making the first payment. That is, determine Bj

Trout Population A pond currently has 2000 trout in it. A fish hatchery decides to add an additional 20 trout each month. In addition, it is known that the trout population is growing 3% per month. The size of the population after n months is given by the recursively defined sequence Po = 2000, Pn = l.03pn-l + 20 How many trout are in the pond after two months? That is, what is P2 ?

Car Loans Phil bought a car by taking out a loan for $18,500 at 0.5% interest per month. Phil's normal monthly payment is $434.47 per month, but he decides that he can afford to pay $100 extra toward the balance each month. His balance each month is given by the recursively defined sequence Bo = $18,500, Bn = 1.005Bn_1 - 534.47 Determine Phil's balance after making the first payment. That is, determine Bl .

Environmental Control The Environmental Protection Agency (EPA) determines that Maple Lake has 250 tons of pollutant as a result of industrial waste and that 10% of the pollutant present is neutralized by solar oxidation every year. The EPA imposes new pollution control laws that result in 15 tons of new pollutant entering the lake each year. The amount of pollutant in the lake after n years is given by the recursively defined sequence Po = 250, Pn = O.9Pn-l + 15 Determine the amount of pollutant in the lake after 2 years. That is, determine P2'

Growth of a Rabbit Colony A colony of rabbits begins with one pair of mature rabbits, which will produce a pair of offspring (one male, one female) each month. Assume that all rabbits mature in 1 month and produce a pair of offspring (one male, one female) after 2 months. If no rabbits 1 2 3 n 66. - + - + - + ... + - e e2 e3 e" 68. a + ar + ar2 + ... + arn-J 40 71. 2:k k = l 16 75. 2: (k2 + 4) k=1 20 79. 2: k3 k=5 24 72. 2: (-k) k= l 14 76. 2: (k2 - 4) k=O 24 80. 2: k3 k=4 ever die, how many pairs of mature rabbits are there after 7 months? [Hint: A Fibonacci sequence models this colony. Do you see why?] 1 mature pair 1 mature pair 2 mature pairs 3 mature pairs

Fibonacci Sequence Let Un = (1 + vs)" - (1 - vs)n 2 nVS define the nth term of a sequence. (a) Show that Uj = 1 and U2 = 1. (b) Show that Un+2 = Un+l + UI1 " ( c) Draw the conclusion that {un} is a Fibonacci sequence.

Pascal's Triangle Divide the triangular array shown (called Pascal's triangle) using diagonal lines as indica ted. Find the sum of the numbers in each diagonal row. Do you recognize this sequence? / ffi, 1 1 5 10 10 5 6 15 20 15 6

Fibonacci Sequence Use the result of Problem 86 to do the following problems: (a) Write the first 11 terms of the Fibonacci sequence. Un+l (b) Write down the first 10 terms of the ratio --. Un (c) As n gets large, what number does the ratio approach? This number is referred to as the golden ratio. Rectangles whose sides are in this ratio were considered pleasing to the eye by the Greeks. For example, the facade of the Parthenon was constructed using the golden ratio. Ull (d) Write down the first 10 terms of the ratio --. UIl+1 (e) As n gets large, what number does the ratio approach? This number is also referred to as the conjugate golden ratio. This ratio is believed to have been used in the construction of the Great Pyramid in Egypt. The ratio equals the sum of the areas of the four face triangles divided by the total surface area of the Great Pyramid.

Approximating I(x) = e ' In calculus, it can be shown that 00 x k [(x) = e X = k=ok! We can approximate the value of [(x) = e X for any x using the following sum for some n. Il :ck [(x) = eX "" k=O k! (a) Approximate [(1.3) with n = 4 (b) Approximate [ (1 .3) with n = 7. (c) Use a calculator to approximate [ (1 .3). ', (d) Using trial and error along with a graphing utility'S SEQuence mode, determine the value of n required to approximate [ (1.3) correct to eight decimal places.

Approximating I(x) = e ' Refer to Problem 89. (a) Approximate [( - 2.4) with n = 3. (b) Approximate [( - 2.4) with n = 6. (c) Use a calculator to approximate [(- 2.4) . : (d) Using trial and error along with a graphing utility' S SEQuence mode, determine the value of n required to approximate [( - 2.4) correct to eight decimal places.

Bode's Law In 1772, Johann Bode published the following formula for predicting the mean distances, in astronomical units (AU), of the planets from the sun: a1 = 0.4, {a,J = {0.4 + 0.3 ' 21l-2} , n 2 where n is the number of the planet from the sun. (a) Determine the first eight terms of this sequence. (b) At the time of Bode's publication, the known planets were Mercury (0.39 AU), Venus (0.72 AU), Earth (1 AU), Mars (1.52 AU), Jupiter (5.20 AU), and Saturn (9.54 AU). How do the actual distances compare to the terms of the sequence? (c) The planet Uranus was discovered in 1781 and the asteroid Ceres was discovered in 1801. The mean orbital distances from the sun to Uranus and Ceres';' are 19.2 AU and 2.77 AU, respectively. How well do these values fit within the sequence? (d) Determine the ninth and tenth terms of Bode's sequence. (e) The planets Neptune and Pluto* were discovered in 1846 and 1930, respectively. Their mean orbital distances from the sun are 30.07 AU and 39.44 AU, respectively. How do these actual distances compare to the terms of the sequence? (f) On July 29, 2005, NASA announced the discovery of a tenth planet* (n = 11), which has temporarily been named 2003 UB313* until a permanent name is decided on. Use Bode's Law to predict the mean orbital distance of 2003 UB313 from the sun. Its actual mean distance is not yet known, but 2003 UB313 is currently about 97 astronomical units from the sun. Sources: NASA.

Show that n(n + 1) 1 + 2 + ... + (n - 1) + n = --2-- [Hint: Let s = 1 + 2 + ... + (n - 1) + n S = n + (n - 1) + (n - 2) + . .. + 1 Add these equations. Then 25 = [1 + nJ + [2 + (n - 1)] + ... + [n + 1] n terms in brackets Now complete the derivation.]

Investigate various applications that lead to a Fibonacci sequence, such as art, architecture, or financial markets. Write an essay on these applications.

In a(n) __ sequence, the difference between successive terms is a constant.

True or False In an arithmetic sequence the sum of the first and last terms equals twice the sum of all the terms.

In Problems 3-12, show that each sequence is arithmetic. Find the common difference and write out the first four terms. (sill = {n + 4}

In Problems 3-12, show that each sequence is arithmetic. Find the common difference and write out the first four terms. (sill = {n - 5 }

In Problems 3-12, show that each sequence is arithmetic. Find the common difference and write out the first four terms. (aliI = {2n - 5 }

In Problems 3-12, show that each sequence is arithmetic. Find the common difference and write out the first four terms. {bill = {3n + I}

In Problems 3-12, show that each sequence is arithmetic. Find the common difference and write out the first four terms. (clll = {6 - 2n}

In Problems 3-12, show that each sequence is arithmetic. Find the common difference and write out the first four terms. (all) = {4 - 2n}

In Problems 3-12, show that each sequence is arithmetic. Find the common difference and write out the first four terms. ( t I = { ! - ! n }

In Problems 3-12, show that each sequence is arithmetic. Find the common difference and write out the first four terms. ( till = 3 + 4

In Problems 3-12, show that each sequence is arithmetic. Find the common difference and write out the first four terms. (sill = {In 3 11}

In Problems 3-12, show that each sequence is arithmetic. Find the common difference and write out the first four terms. (sill = {e 1nll }

In Problems 13-20, find the nth term of the arithmetic sequence (alII whose initial term a and common difference d are given. What is the fifty-first term? al = 2; d = 3

In Problems 13-20, find the nth term of the arithmetic sequence (alII whose initial term a and common difference d are given. What is the fifty-first term? al = -2; d = 4

In Problems 13-20, find the nth term of the arithmetic sequence (alII whose initial term a and common difference d are given. What is the fifty-first term? a1 = 5; d = -3

In Problems 13-20, find the nth term of the arithmetic sequence (alII whose initial term a and common difference d are given. What is the fifty-first term? a1 = 6; d = -2

In Problems 13-20, find the nth term of the arithmetic sequence (alII whose initial term a and common difference d are given. What is the fifty-first term? al = O' , d = -2

In Problems 13-20, find the nth term of the arithmetic sequence (alII whose initial term a and common difference d are given. What is the fifty-first term? al = 1; d = -3

In Problems 13-20, find the nth term of the arithmetic sequence (alII whose initial term a and common difference d are given. What is the fifty-first term? a1 = v2; d = v

In Problems 13-20, find the nth term of the arithmetic sequence (alII whose initial term a and common difference d are given. What is the fifty-first term? aJ = 0; d = 7T

In Problems 21-26, find the indicated term in each arithmetic sequence. 100th term of 2, 4, 6, ...

In Problems 21-26, find the indicated term in each arithmetic sequence. 80th term of -1, 1, 3, ...

In Problems 21-26, find the indicated term in each arithmetic sequence. 90th term of 1, -2, -5, ...

In Problems 21-26, find the indicated term in each arithmetic sequence. 80th term of 5, 0, -5, ...

In Problems 21-26, find the indicated term in each arithmetic sequence. 80th term of 2, 2' 3, 2' . . .

In Problems 21-26, find the indicated term in each arithmetic sequence. 70th term of 2Vs, 4Vs, 6Vs, . . .

In Problems 27-34, find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term. 8th term is 8; 20th term is 44

In Problems 27-34, find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term. 4th term is 3; 20th term is 35

In Problems 27-34, find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term. 9th term is -5; 15th term is 31

In Problems 27-34, find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term. 8th term is 4; 1 8th term is -96

In Problems 27-34, find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term. 15th term is 0; 40th term is -50

In Problems 27-34, find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term. 5th term is -2; 13th term is 30

In Problems 27-34, find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term. 14th term is -1; 18th term is -9

In Problems 27-34, find the first term and the common difference of the arithmetic sequence described. Give a recursive formula for the sequence. Find a formula for the nth term. 12th term is 4; 1 8th term is 28

In Problems 35-52, find each sum. 1 + 3 + 5 + . . . + (2n - 1)

In Problems 35-52, find each sum. 2 + 4 + 6 + ... + 2n

In Problems 35-52, find each sum. 7 + 12 + 17 + ... + (2 + 5n)

In Problems 35-52, find each sum. -1 + 3 + 7 + . . . + (4n - 5)

In Problems 35-52, find each sum. 2 + 4 + 6 + ... + 70

In Problems 35-52, find each sum. 1 + 3 + 5 + . . . + 59

In Problems 35-52, find each sum. 5 + 9 + 13 + ... + 49

In Problems 35-52, find each sum. 2 + 5 + 8 + ... + 41

In Problems 35-52, find each sum. 73 + 78 + 83 + 88 + ... + 558

In Problems 35-52, find each sum. 7+ 1 - 5 - 1 1 -- 299

In Problems 35-52, find each sum. 4 + 4.5 + 5 + 5.5 + ... + 100

In Problems 35-52, find each sum. 8 + 8- + 8- + 8- + 9 + ... + 50 424

In Problems 35-52, find each sum. 80 2:(2n - 5)

In Problems 35-52, find each sum. 90 2: (3 - 2n) 11 =1

In Problems 35-52, find each sum. JOO ( 1 ) 2: 6 - - n 11=1 2

In Problems 35-52, find each sum. 80 ( 1 1 ) 2: -n + - 11 =1 3 2

In Problems 35-52, find each sum. The sum of the first 120 terms of the sequence 14, 16, 18, 20, ...

In Problems 35-52, find each sum. The sum of the first 46 terms of the sequence 2, -1, -4, -7, . . . .

Find x so that x + 3, 2x + 1, and 5x + 2 are consecutive terms of an arithmetic sequence.

Find x so that 2x, 3x + 2, and 5x + 3 are consecutive terms of an arithmetic sequence.

Drury Lane Theater The Drury Lane Theater has 25 seats in the first row and 30 rows in all. Each successive row contains one additional seat. How many seats are in the thea ter?

Football Stadium The corner section of a football stadium has 15 seats in the first row and 40 rows in all. Each successive row contains two additional seats. How many seats are in this section?

Creating a Mosaic A mosaic is designed in the shape of an equilateral triangle, 20 feet on each side. Each tile in the mosaic is in the shape of an equilateral triangle, 12 inches to a side. The tiles are to alternate in color as shown in the illustration. How many tiles of each color will be required? 20 ' / T 'II' '... . ... ' , 20 ' TTTTTT T 'T T T ... T ... T ... T TT TT TTTT 20 '

Constructing a Brick Staircase A brick staircase has a total of 30 steps. The bottom step requires 100 bricks. Each successive step requires two less bricks than the prior step. (a) How many bricks are required for the top step? (b) How many bricks are required to build the staircase?

Cooling Air As a parcel of air rises (for example, as it is pushed over a mountain), it cools at the dry adiabatic lapse rate of 5.5 F per 1000 feet until it reaches its dew point. If the ground temperature is 67 F, write a formula for the sequence of temperatures, {Til), of a parcel of air that has risen n. thousand feet. What is the temperature of a parcel of air if it has risen 5000 feet? Source: National Aeronautics and Space Administration

Citrus Ladders Ladders used by fruit pickers are typically tapered with a wide bottom for stability and a narrow top for ease of picking. Suppose the bottom rung of such a ladder is 49 inches wide and the top rung is 24 inches wide. How many rungs does the ladder have if each rung is 2.5 inches shorter than the one below it? How much material would be needed to make the rungs for the ladder described? Source: www.slokesladclers.com

Seats in an Aml )hitheater An outdoor amphitheater has 35 seats in the first row, 37 in the second row, 39 in the third row, and so on. There are 27 rows altogether. How many can the amphitheater seat?

Stadium Construction How many rows are in the corner section of a stadium containing 2040 seats if the first row has 10 seats and each successive row has 4 additional seats?

Salary Suppose that you j ust received a job offer with a starting salary of $35,000 per year and a guaranteed raise of $1400 per year. How many years will it take before your aggregate salary is $280,000? [Hint: Your aggregate salary after 2 years is $35,000 + ($35,000 + $1400).]

Make up an arithmetic sequence. Give it to a friend and ask for its twentieth term.

Describe the similarities and differences between arithmetic sequences and linear functions.

If $1000 is invested at 4% per annum compounded semiannually, how m uch is in the account after two years? (pp. 465-472)

How much do you need to invest now at 5% per annum compounded monthly so that in 1 year you will have $10, 000? (pp. 465-472)

In a(n) ____ sequence the ratio of successive terms is a constant.

If 11'1 < 1, the sum of the geometric series 2: ark-I k=1 is ___.

A sequence of equal periodic deposits is called a(n) ____.

True or False A geometric sequence may be defined recursively.

True or False In a geometric sequence the common ratio is always a positive number.

True or False For a geometric sequence with first term al and common ratio 1', where I' of. 0, I' of. 1, the sum of the first 1 - I'" n terms is SI1 = al '-- . 1 - I'

In Problems 9-18, a geometric sequence is given. Find the common ratio and write out the first four terms. {Sill = {3"}

In Problems 9-18, a geometric sequence is given. Find the common ratio and write out the first four terms. {Sill = {( -5)"}

In Problems 9-18, a geometric sequence is given. Find the common ratio and write out the first four terms. a"l = { -3G)"}

In Problems 9-18, a geometric sequence is given. Find the common ratio and write out the first four terms. bill = {G),,}

In Problems 9-18, a geometric sequence is given. Find the common ratio and write out the first four terms. {clll = -4

In Problems 9-18, a geometric sequence is given. Find the common ratio and write out the first four terms. {dill = 9

In Problems 9-18, a geometric sequence is given. Find the common ratio and write out the first four terms. {e"l = {21 l /3}

In Problems 9-18, a geometric sequence is given. Find the common ratio and write out the first four terms. {fnl = {321 1 }

In Problems 9-18, a geometric sequence is given. Find the common ratio and write out the first four terms. { tl1l =

In Problems 9-18, a geometric sequence is given. Find the common ratio and write out the first four terms. unl = 3'I - l

In Problems 19-32, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. {n + 2}

In Problems 19-32, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. {2n - 5}

In Problems 19-32, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. {4n2 }

In Problems 19-32, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. {5n2 + I}

In Problems 19-32, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. { 3 - n}

In Problems 19-32, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. { 8 - n }

In Problems 19-32, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. 1 , 3, 6, 10, ...

In Problems 19-32, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. 2, 4, 6, 8, ...

In Problems 19-32, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. { ()"}

In Problems 19-32, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. {(%)"}

In Problems 19-32, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. -1, -2, -4, -8, ...

In Problems 19-32, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. 1 , 1, 2, 3, 5, 8, ...

In Problems 19-32, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. {3n/ 2}

In Problems 19-32, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. {( -I)"}

In Problems 33-40, find the fifth term and the nth term of the geometric sequence whose initial term a1 and common ratio r are given. al = 2; 1' = 3

In Problems 33-40, find the fifth term and the nth term of the geometric sequence whose initial term a1 and common ratio r are given. a1 = -2; 1'=4

In Problems 33-40, find the fifth term and the nth term of the geometric sequence whose initial term a1 and common ratio r are given. a1 = 5; 1' = -1

In Problems 33-40, find the fifth term and the nth term of the geometric sequence whose initial term a1 and common ratio r are given. al = 6; 1'=-2

In Problems 33-40, find the fifth term and the nth term of the geometric sequence whose initial term a1 and common ratio r are given. al = 0; r = .

In Problems 33-40, find the fifth term and the nth term of the geometric sequence whose initial term a1 and common ratio r are given. al = 1; r=- .

In Problems 33-40, find the fifth term and the nth term of the geometric sequence whose initial term a1 and common ratio r are given. al = V2; r = V2

In Problems 33-40, find the fifth term and the nth term of the geometric sequence whose initial term a1 and common ratio r are given. al = 0; 1' = 1. .

In Problems 41-46, find the indicated term of each geometric sequence. 7th term of 1'"2'"4 ""

In Problems 41-46, find the indicated term of each geometric sequence. 8th term of 1 , 3, 9, . . .

In Problems 41-46, find the indicated term of each geometric sequence. 9th term of 1, -1, 1, .. .

In Problems 41-46, find the indicated term of each geometric sequence. 10th term of -1,2, -4, ...

In Problems 41-46, find the indicated term of each geometric sequence. 8th term of 0.4, 0.04, 0.004, ...

In Problems 41-46, find the indicated term of each geometric sequence. 7th term of 0.1, 1. 0, 10.0, ...

In Problems 47-54, find the nth term all of each geometric sequence. When given, r is the common ratio. 7, 14, 28, 56, ...

In Problems 47-54, find the nth term all of each geometric sequence. When given, r is the common ratio. 5, 1 0, 20, 40, ...

In Problems 47-54, find the nth term all of each geometric sequence. When given, r is the common ratio. -3,1, -3'9"

In Problems 47-54, find the nth term all of each geometric sequence. When given, r is the common ratio. 4,1, -4'16"

In Problems 47-54, find the nth term all of each geometric sequence. When given, r is the common ratio. a6 = 243; I' = -3

In Problems 47-54, find the nth term all of each geometric sequence. When given, r is the common ratio. a2 = 7; I' =-3

In Problems 47-54, find the nth term all of each geometric sequence. When given, r is the common ratio. (/2 = 7; a4 = 1575

In Problems 47-54, find the nth term all of each geometric sequence. When given, r is the common ratio. a3 = 1; a6 = 1

In Problems 55-60, find each sum. 1 2 22 2 3 2 11-1 - + -+ -+ - + . . 4 4 4 4 .. + --4

In Problems 55-60, find each sum. 3 32 33 3 11 -+ - + - + ... + - 9 9 9 9

In Problems 55-60, find each sum. i ()k k=1 3

In Problems 55-60, find each sum. 2.: 4 3k-1 k=l

In Problems 55-60, find each sum. -1 -2 -4 -8 - ... - (2"-1)

In Problems 55-60, find each sum. 2 + - + - + ... + 2 - 5 25 5

For Problems 61-66, use a graphing utility to find the sum of each geometric sequence. 1 2 22 2 3 2 14 3 -+ -+ - + - + .. . + - 62. -+ - + -+ . .. + - 4 4 44 4

For Problems 61-66, use a graphing utility to find the sum of each geometric sequence. 3 32 3 3 315 -+ -+ - + - + .. . + - 9 9 9 9

For Problems 61-66, use a graphing utility to find the sum of each geometric sequence. 5 ( 2 )11 2.: - 11=1 3

For Problems 61-66, use a graphing utility to find the sum of each geometric sequence. 15 2.: 4 .3 11-1

For Problems 61-66, use a graphing utility to find the sum of each geometric sequence. -1 - 2 - 4 - 8 - ... - 2 14

For Problems 61-66, use a graphing utility to find the sum of each geometric sequence. 2 + -+ - + . .. + 2 - 5 25 5

In Problems 67-82, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 1 + -+ -+ ... 3 9

In Problems 67-82, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 2 + -+ - + ... 3 9

In Problems 67-82, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 8 + 4 + 2 + ...

In Problems 67-82, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 6 + 2 + 3 + ...

In Problems 67-82, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 2 --+ - --+ ... 2 8 32

In Problems 67-82, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 1 --+---+ 4 16 64

In Problems 67-82, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 8 + 12 + 18 + 27 + ...

In Problems 67-82, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 9 + 12 + 16 + 3 + ...

In Problems 67-82, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 00 (l)k-l 2.: 5 - k=1

In Problems 67-82, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 00 (l)k-l 2.: 8 - k=l 3

In Problems 67-82, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 00 1 2.: - 3 k-1 k=12

In Problems 67-82, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 00 (3)k-1 2.: 3 - k=1 2

In Problems 67-82, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 2.: 6 - - k=1 3

In Problems 67-82, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 00 ( l)k-1 2.: 4 -- k=l 2

In Problems 67-82, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 00 (2)k 3

In Problems 67-82, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 2.:2 - k=l 4

Find x so that x, x + 2, and x + 3 are consecutive terms of a geometric sequence.

Find x so that x - 1, x, and x + 2 are consecutive terms of a geometric sequence.

Salary Increases Suppose that you have j ust been hired at an annual salary of $18,000 and expect to receive annual increases of 5%. What will your salary be when you begin your fifth year?

Equipment Depreciation A new piece of equipment cost a company $15,000. Each year, for tax purposes, the company depreciates the value by 15%. What value should the company give the equipment after 5 years?

Pendulum Swings Initially, a pendulum swings through an arc of 2 feet. On each successive swing, the length of the arc is 0.9 of the previous length. (a) What is the length of the arc of the 10th swing? (b) On which swing is the length of the arc first less than 1 foot? (c) After 15 swings, what total length will the pendulum have swung? (d) When it stops, what total length will the pendulum have swung?

Bouncing Balls A ball is dropped from a height of 30 feet. Each time it strikes the ground, it bounces up to 0.8 of the previous height. -1 1 /\ 0 / I 1\ / I /1 30' I 1 \ 1 \ 1\ 1 \ / 24' /19.2'1 \ ?t I / I / I / I /1 I / I / I / I / I 1/ I / 1/ 1/1 1/ 1/ 1/ 1/ I 1/ 1/ 1/ 1/ I 1/ 1/ I (a) What height will the ball bounce up to after it strikes the ground for the third time? (b) How high will it bounce after it strikes the ground for the nth time? (c) How many times does the ball need to strike the ground before its bounce is less than 6 inches? (d) What total distance does the ball travel before it stops bouncing?

Retirement Christine contributes $100 each month to her 401(k). What will be the value of Christine's 401(k) after the 360th deposit (30 years) if the per annum rate of return is assumed to be 12% compounded monthly?

SaYing for a Home Jolene wants to purchase a new home. Suppose that she invests $400 per month into a mutual fund. If the per annum rate of return of the mutual fund is assumed to be 10% compounded monthly, how much will Jolene have for a down payment after the 36th deposit (3 years)?

Tax Sheltered Annuity Don contributes $500 at the end of each quarter to a tax sheltered annuity (TSA). What will the value of the TSA be after the 80th deposit (20 years) if the per annum rate of return is assumed to be 8% compounded quarterly?

Retirement Ray contributes $1 000 to an Individual Retirement Account (IRA) semiannually. What will the value of the IRA be when Ray makes his 30th deposit (after 15 years) if the per annum rate of return is assumed to be 10% compounded semiannually?

Sinking Fund Scott and Alice want to purchase a vacation home in 10 years and need $50,000 for a down payment. How much should they place in a savings account each month if the per annum rate of return is assumed to be 6% compounded monthly?

Sinking Fund For a child born in 1 996, a 4-year college education at a public university is projected to be $150,000. Asslllning an 8% per annum rate of return compounded monthly, how much must be contributed to a college fund every month to have $150,000 in 18 years when the child begins college?

Grains of Wheat on a Chess Board In an old fable, a commoner who had saved the king's life was told he could ask the king for any just reward. Being a shrewd man, the commoner said, "A simple wish, sire. Place one grain of wheat on the first square of a chessboard, two grains on the second square, four grains on the third square, continuing until you have filled the board. This is all I seek." Compute the total number of grains needed to do this to see why the request, seemingly simple, could not be granted. (A chessboard consists of 8 X 8 = 64 squares.)

Look at the figure. What fraction of the square is eventually shaded if the indicated shading process continues indefinitely?

Multiplier Suppose that, throughout the U.S. economy, individuals spend 90% of every additional dollar that they earn. Economists would say that an individual's marginal propensity to consume is 0.90. For example, if Jane earns an additional dollar, she will spend 0.9(1) = $0. 90 of it. The individual that earns $0.90 (from Jane) will spend 90% of it or $0.81. This process of spending continues and results in an infinite geometric series as follows: 1, 0.90, 0. 902 , 0.903 , 0.904 , ... The sum of this infinite geometric series is called the multiplier. What is the multiplier if individuals spend 90% of every additional dollar that they earn?

Multiplier Refer to Problem 97. Suppose that the marginal propensity to consume throughout the U.S. economy is 0.95. What is the multiplier for the U.S. economy?

Stock Price One method of pricing a stock is to discount the stream of future dividends of the stock. Suppose that a stock pays $P per year in dividends and, historically, the dividend has been increased i% per year. If you desire an annual rate of return of r%, this method of pricing a stock states that the price that you should pay is the present value of an infinite stream of payments: 1 + i (1 + i)2 ( 1 + i ) 3 Price = P + P--+ P -- + P -- + ... l+r 1+, . l +r The price of the stock is the sum of an infinite geometric series. Suppose that a stock pays an annual dividend of $4.00 and, historically, the dividend has been increased 3% per year. You desire an annual rate of return of 9%. What is the most you should pay for the stock?

Stock Price Refer to Problem 99. Suppose that a stock pays an annual dividend of $2.50 and, historically, the dividend has increased 4 % per year. You desire an annual rate of return of 11 %. What is the most that you should pay for the stock?

A Rich Man's Promise A rich man promises to give you $1000 on September 1, 2007. Each day thereafter he will give you :0 of what he gave you the previous day. What is the first date on which the amount you receive is less than 1? How much have you received when this happens?

Critical Thinking You are interviewing for a job and receive two offers: A: $20,000 to start, with guaranteed annual increases of 6% for the first 5 years B: $22,000 to start, with guaranteed annual increases of 3% for the first 5 years Which offer is best if your goal is to be making as much as possible after 5 years? Which is best if your goal is to make as much money as possible over the contract (5 years)?

Critical Thinking Which of the following choices, A or B, results in more money? A: To receive $1000 on day 1, $ 999 on day 2, $ 998 onday 3, with the process to end after 1000 days B: To receive $1 on day 1, $2 on day 2, $4 on day 3, for 19 days

Critical Thinking You have just signed a 7-year professional football league contract with a beginning salary of $2,000,000 per year. Management gives you the following options with regard to your salary over the 7 years. 1. A bonus of $100,000 each year 2. An annual increase of 4.5% per year beginning after 1 year 3. An annual increase of $ 95,000 per year beginning after 1 year Which option provides the most money over the 7-year period? Which the least? Which would you choose? Why?

Critical Thinking Suppose you were offered a job in which you would work 8 hours per day for 5 workdays per week for 1 month at hard manual labor. Your pay the first day would be 1 penny. On the second day your pay would be two pennies; the third day 4 pennies. Your pay would double on each successive workday. There are 22 workdays in the month. There will be no sick days. If you miss a day of work, there is no pay or pay increase. How much would you get paid if you work all 22 days? How much do you get paid for the 22nd workday? What risks do you run if you take this job offer? Would you take the job?

Can a sequence be both arithmetic and geometric? Give reasons for your answer.

Make up a geometric sequence. Give it to a friend and ask for its 20th term.

Make up two infinite geometric series, one that has a sum and one that does not. Give them to a friend and ask for the sum of each series.

Describe the similarities and differences between geometric sequences and exponential functions.

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 2 + 4 + 6 + .. . + 2n = n(n + 1)

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 1 + 5 + 9 + . . . + (4n - 3) = n(2n - 1)

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 3 + 4 + 5 + ... + (n + 2) = 2 n(n + 5)

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 3 + 5 + 7 + . .. + (2n + 1) = n(n + 2)

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 2 + 5 + 8 + ... + (3n - 1) = 2 n(3n + 1)

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 1 + 4 + 7 + ... + (3n - 2) = 2n(3n - 1)

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 1 + 2 + 22 + ... + 2//-1 = 2// - 1

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 1 + 3 + 32 + . .. + 3n-1 = .!.(3// - 1) 2

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 1 + 4 + 42 + . .. + 4//-1 = (4// - 1)

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 1 + 5 + 52 + ... + 5//-1 = (5n - 1)

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 1 1 1 1 n - + - + - + . . . + --,------,- 1 2 23 34 n(n + l ) n+l

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 1 1 1 1 n - + - + - + . . . + ------- 13 35 5 7 (2n - 1) (2n + 1) 2n + 1

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 1 + 2-+ 3- + ... + n = "6 n(n + 1 ) (2n + 1)

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 13 + 23 + 33 + ... + n3 = "4 n2(n + 1)2

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 4 + 3 + 2 + ... + (5 - n) = 2n(9 - n)

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. -2 - 3 - 4 - ... - (n + 1) = -2n(n + 3)

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 1 2 + 23 + 34 + ... + n(n + 1) = n(n + l)(n + 2)

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 1,2 + 34 + 56 + . .. + (2n - 1)(2n) = n(n + 1)(4n - 1)

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. n2 + n is divisible by 2.

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. n3 + 2n is divisible by 3.

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. n2 - n + 2 is divisible by 2.

In Problems 1-22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. n(n + 1)(n + 2) is divisible by 6.

In Problems 23-27, prove each statement. If x > 1, then xn > 1.

In Problems 23-27, prove each statement. If 0 < x < 1, then 0 < xn < 1.

In Problems 23-27, prove each statement. a - b is a factor of an - bn. [Hint: ak+1 - bk+1 = a(ak - bk) + b k (a - b)]

In Problems 23-27, prove each statement. a + b is a factor of a2n+ 1 + b2/ /+I.

In Problems 23-27, prove each statement. (1 + a)" 1 + na, for a > 0

Show that the statement " n2 - n + 41 is a prime number" is true for n = 1, but is not true for n = 41.

Show that the formula 2 + 4 + 6 + . . . + 2n = n2 + n + 2 obeys Condition II of the Principle of Mathematical Induction. That is, show that if the formula is true for some k it is also true for k + 1. Then show that the formula is false for n = 1 (or for any other choice of n).

Use mathematical induction to prove that if r =f. 1 then 1 - r// a + ar + ar2 + ... + arn-1 = a

Use mathematical induction to prove that a + (a + d) + (a + 2d) I - r n(n - 1) + ... + [a + (n - l)d] = na + d--'-- 2

Extended Principle of Mathematical Induction The Extended Principle of Mathematical Induction states that if Conditions I and II hold, that is, (I) A statement is true for a natural number j. (II) If the statement is true for some natural number k j, then it is also true for the next natural number k+1. then the statement is true for all natural numbers "2 j. show that the number of diagonals in a convex polygon of n sides IS "2 n n ( - 3 . ) [Hint: Begin by showing that the result is true when n = 4 (Condition I).]

Geometry Use the Extended Principle of Mathematical Induction to show that the sum of the interior angles of a convex polygon of n sides equals (n - 2) . 180.

The ____ is a triangular display of the binomial coefficients.

() =___

True or False ( n ) = ( p) J n - J ! n!

The __ __ can be used to expand expressions like (2x + 3 )6.

In Problems 5-16, evaluate each expression. C)

In Problems 5-16, evaluate each expression. C)

In Problems 5-16, evaluate each expression. G)

In Problems 5-16, evaluate each expression. ()

In Problems 5-16, evaluate each expression. !)

In Problems 5-16, evaluate each expression. ( 1 98)

In Problems 5-16, evaluate each expression. ( 1000 ) 1000

In Problems 5-16, evaluate each expression. ( 1000 )

In Problems 5-16, evaluate each expression. CD

In Problems 5-16, evaluate each expression. G)

In Problems 5-16, evaluate each expression. G)

In Problems 5-16, evaluate each expression. ()

In Problems 17-28, expand each expression using the Binomial Theorem. (x + 1)5

In Problems 17-28, expand each expression using the Binomial Theorem. (x - 1)5

In Problems 17-28, expand each expression using the Binomial Theorem. (x - 2)6

In Problems 17-28, expand each expression using the Binomial Theorem. (x + 3)5

In Problems 17-28, expand each expression using the Binomial Theorem. (3x + 1)4

In Problems 17-28, expand each expression using the Binomial Theorem. (2x + 3)5

In Problems 17-28, expand each expression using the Binomial Theorem. (x2 + l)5

In Problems 17-28, expand each expression using the Binomial Theorem. (x2 - l)6

In Problems 17-28, expand each expression using the Binomial Theorem. (\IX + \12)6

In Problems 17-28, expand each expression using the Binomial Theorem. (\IX - v'3t

In Problems 17-28, expand each expression using the Binomial Theorem. (ax + by)5

In Problems 17-28, expand each expression using the Binomial Theorem. (ax - by)4

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coefficient of x6 in the expansion of (x + 3)10

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coefficient of x 3 in the expansion of (x - 3)10

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coefficient of x 7 in the expansion of (2x - 1)12

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coefficient of x 3 in the expansion of (2x + 1 ) 12

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coefficient of x 7 in the expansion of (2x + 3)9

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coefficient of x 2 in the expansion of (2x - 3)9

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The fifth term in the expansion of (x + 3)

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The third term in the expansion of (x - 3)7

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The third term in the expansion of (3x - 2)9

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The sixth term in the expansion of (3x + 2)8

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coefficient of xo in the expansion of ( x 2 + ) 12

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coef Clent of x Jl1 the expansIOn of x - )9 x 2

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coefficient of X4 in the expansion of ( x _ yo

In Problems 29-42, use the Binomial Theorem to find the indicated coefficient or term. The coefficient of x 2 in the expansion of ( Vx + :rx y

Use the Binomial Theorem to find the numerical value of (1.001)5 correct to five decimal places. [Hint: (1.001)5 = (1 + 10- 3 ) 5 ]

Use the Binomial Theorem to find the numerical value of (0.998)6 correct to five decimal places.

Show that C : 1 ) = n and (: ) = 1.

Show that if n and j are integers with a j n then Conclude that the Pascal triangle is symmetric with respect to a vertical line drawn from the topmost entry.

If n is a positive integer, show that [Hint: 21/ = (1 + 1)"; now use the Binomial Theorem.]

If n is a positive integer, show that (z) - () + G) -. .. + (-1)''(:) = a

()GY + G)()4G) + G)(YGY + C)GYGY + C)G)GY + G)(Y = ?

Stirling's Formula An approximation for n!, when n is large, is given by n! v:2;;;;(!2)n (l + _ 1_ ) e 12n - 1 Calculate 12!, 20!, and 25! on your calculator. Then use Stirling's formula to approximate 12!, 20!, and 25!.

In Problems 1-8, write down the first five terms of each sequence. (all) = {( -1)" (: : D}

In Problems 1-8, write down the first five terms of each sequence. (bll) = {( -1)1l+ 1 (2n + 3)}

In Problems 1-8, write down the first five terms of each sequence. (cll) = """1 n

In Problems 1-8, write down the first five terms of each sequence. (dn) = -

In Problems 1-8, write down the first five terms of each sequence. (dn) = -

In Problems 1-8, write down the first five terms of each sequence. al = 4;

In Problems 1-8, write down the first five terms of each sequence. al = 2; all = 2 - an-l

In Problems 1-8, write down the first five terms of each sequence. al = -3; a n = 4 + an-l

In Problems 9 and 10, write out each sum. 2,: (4k+2) k=i

In Problems 9 and 10, write out each sum. 3 2,: (3 - k2) k=i

In Problems 11 and 12, express each sum using summation notation. 1 --+ 2 --3 4 13

In Problems 11 and 12, express each sum using summation notation. 2 +-3 +-32 + + --3"

In Problems 13-24, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms. (all ) = {n + 5}

In Problems 13-24, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms. (bn) = {4n + 3}

In Problems 13-24, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms. (cll) = {2n 3}

In Problems 13-24, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms. dll) = {2n2 - 1}

In Problems 13-24, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms. (sn) = (23n)

In Problems 13-24, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms. (un) = (23n)

In Problems 13-24, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms. 0, 4, 8, 12, ...

In Problems 13-24, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms. 1, -3, -7, -11, ...

In Problems 13-24, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms. 3 3 3 3 3 "2'"4'8'16' ...

In Problems 13-24, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms. 5 5 5 5 5 "3'"9'27'81' ...

In Problems 13-24, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms.2 3 4 5 "3'"4'5'6' ...

In Problems 13-24, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the sequence is geometric, find the common ratio and the sum of the first n terms. 3 5 7 9 11 "2'"4'6'8'10 ...

In Problems 25-30, find each sum. 50 (3k) k=l

In Problems 25-30, find each sum. 30 k2 k=l

In Problems 25-30, find each sum. 30 (3k - 9) k=l

In Problems 25-30, find each sum.4 0 ( -2k + 8) k=l

In Problems 25-30, find each sum. 7 (l )k -::;- k=\ -

In Problems 25-30, find each sum. 10 (_2)k k=l

In Problems 31-36, find the indicated term in each sequence. [Hint: Find the general term first.] 9th term of 3, 7,11,15,...

In Problems 31-36, find the indicated term in each sequence. [Hint: Find the general term first.] 8th term of 1, -1, -3, -5,...

In Problems 31-36, find the indicated term in each sequence. [Hint: Find the general term first.] 11th term of 1, 10' 100""

In Problems 31-36, find the indicated term in each sequence. [Hint: Find the general term first.] 11th term of 1, 4, 2, 8, ...

In Problems 31-36, find the indicated term in each sequence. [Hint: Find the general term first.] 9th term of Yz, 2Yz, 3Yz, ...

In Problems 31-36, find the indicated term in each sequence. [Hint: Find the general term first.] 9th term of V 2 , 2, 20 , ..

In Problems 37-40, find a general formula for each arithmetic sequence. 7th term is 31; 20th term is 96

In Problems 37-40, find a general formula for each arithmetic sequence. 8th term is -20; 17th term is -47

In Problems 37-40, find a general formula for each arithmetic sequence. 10th term is 0; 18th term is 8

In Problems 37-40, find a general formula for each arithmetic sequence. 12th term is 30; 22nd term is 50

In Problems 41-48, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 3 + 1 + "3 + "9 + . . .

In Problems 41-48, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 2 + 1 + "2 + 4" + . . .

In Problems 41-48, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 2 - 1 + "2 - 4" + . . .

In Problems 41-48, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 6 -4 + "3 - 9 + ...

In Problems 41-48, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 1 3 9 -+ - + -+ . .. 24 8

In Problems 41-48, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 00 (5 )k-l

In Problems 41-48, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. l)k-l

In Problems 41-48, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 00 ( 3 )k-l 3 -- k=l 4

In Problems 49-54, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers. 3 + 6 + 9 + ... + 3n = 2 (n + 1)

In Problems 49-54, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers. 2 + 6 + 10 + ... + (4n - 2) = 2112

In Problems 49-54, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers. 2 + 6 + 18 + ... + 2.311-J = 311 - 1

In Problems 49-54, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers. 3 + 6 + 12 + . .. + 3.211-1 = 3(2/1 - 1)

In Problems 49-54, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers. 12 + 42 + 72 + ... + (n - 2)2 = 1n (6n2 -3n - 1)

In Problems 49-54, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers. 13 + 24 + 3 5 + ... + n(n + 2) = 6(11 + 1 ) (2n + 7)

In Problems 55 and 56, evaluate each binomial coefficient. G)

In Problems 55 and 56, evaluate each binomial coefficient. G)

In Problems 57-60, expand each expression using the Binomial Theorem. (x + 2)5

In Problems 57-60, expand each expression using the Binomial Theorem. (x - 3)4

In Problems 57-60, expand each expression using the Binomial Theorem. (2x + 3)5

In Problems 57-60, expand each expression using the Binomial Theorem. (3x 4)4

Find the coefficient of x7 in the expansion of (x + 2t

Find the coefficient of x3 in the expansion of (x - 3)8.

Find the coefficient of x 2 in the expansion of (2x + 1)7.

Find the coefficient of x 6 in the expansion of (2x + 1 )8.

Constructing a Brick Staircase A brick staircase has a total of 25 steps. The bottom step requires 80 bricks. Each successive step requires three less bricks than the prior step. (a) How many bricks are required for the top step? (b) How many bricks are required to build the staircase?

Creating a Floor Design A mosaic tile floor is designed in the shape of a trapezoid 30 feet wide at the base and 15 feet wide at the top. The tiles, 12 inches by 12 inches, are to be placed so that each successive row contains one less tile than the row below. How many tiles will be required?

Bouncing Balls A ball is dropped from a height of 20 feet. Each time it strikes the ground, it bounces up to threequarters of the previous height. (a) What height will the ball bounce up to after it strikes the ground for the third time? (b) How high will it bounce after it strikes the ground for the nth time? (c) How many times does the ball need to strike the ground before its bounce is less than 6 inches? (d) What total distance does the ball travel before it stops bouncing?

Retirement Planning Chris gets paid once a month and contributes $200 each pay period into 401(k). his If Chris plans on retiring in 20 years, what will be the value of 401(k) his if the per annum rate of return of the 401(k) is 10% compounded monthly?

Retirement Planning Jacky contributes $500 every quarter to an IRA. If Jacky plans on retiring in 30 years, what will be the value of the IRA if the per annum rate of return of the IRA is 8% compounded quarterly?

Salary Increases Your friend has just been hired at an annual salary of $20,000. If she expects to receive annual increases 4%, of what will be her salary as she begins her fifth year?