apply sigma notation to write the sum.x + x2 + x3 2 + x4 6

write the rst four terms of the sequence. Assume n starts at 1.

write the rst four terms of the sequence. Assume n starts at 1.

write the rst four terms of the sequence. Assume n starts at 1.

write the rst four terms of the sequence. Assume n starts at 1.

write the rst four terms of the sequence. Assume n starts at 1.

write the rst four terms of the sequence. Assume n starts at 1.

write the rst four terms of the sequence. Assume n starts at 1.

write the rst four terms of the sequence. Assume n starts at 1.

write the rst four terms of the sequence. Assume n starts at 1.

write the rst four terms of the sequence. Assume n starts at 1.

write the rst four terms of the sequence. Assume n starts at 1.

write the rst four terms of the sequence. Assume n starts at 1.

nd the indicated term of the sequence.

nd the indicated term of the sequence.

nd the indicated term of the sequence.

nd the indicated term of the sequence.

nd the indicated term of the sequence.

nd the indicated term of the sequence.

nd the indicated term of the sequence.

nd the indicated term of the sequence.

write an expression for the nth term of the given sequence.2, 4, 6, 8, 10, . . .

write an expression for the nth term of the given sequence. 3, 6, 9, 12, 15, . . .

write an expression for the nth term of the given sequence.

write an expression for the nth term of the given sequence.1 2 , 1 4 , 1 8 , 1 16 , 1 32 ,

write an expression for the nth term of the given sequence.

write an expression for the nth term of the given sequence.

write an expression for the nth term of the given sequence.

write an expression for the nth term of the given sequence.1 3 , 2 4 , 3 5 , 4 6 , 5 7 , 1

simplify the ratio of factorials.

simplify the ratio of factorials.

simplify the ratio of factorials.

simplify the ratio of factorials.

simplify the ratio of factorials.

simplify the ratio of factorials.

simplify the ratio of factorials.

simplify the ratio of factorials.

simplify the ratio of factorials.

simplify the ratio of factorials.

simplify the ratio of factorials.

simplify the ratio of factorials.

write the rst four terms of the sequence dened by the recursion formula. Assume the sequence begins at 1.

write the rst four terms of the sequence dened by the recursion formula. Assume the sequence begins at 1.

write the rst four terms of the sequence dened by the recursion formula. Assume the sequence begins at 1.

write the rst four terms of the sequence dened by the recursion formula. Assume the sequence begins at 1.

write the rst four terms of the sequence dened by the recursion formula. Assume the sequence begins at 1.

write the rst four terms of the sequence dened by the recursion formula. Assume the sequence begins at 1.

write the rst four terms of the sequence dened by the recursion formula. Assume the sequence begins at 1.

write the rst four terms of the sequence dened by the recursion formula. Assume the sequence begins at 1.

write the rst four terms of the sequence dened by the recursion formula. Assume the sequence begins at 1.

write the rst four terms of the sequence dened by the recursion formula. Assume the sequence begins at 1.

evaluate the nite series.

evaluate the nite series.

evaluate the nite series.

evaluate the nite series.

evaluate the nite series.

evaluate the nite series.

evaluate the nite series.

evaluate the nite series.

evaluate the nite series.

evaluate the nite series.

evaluate the nite series.

evaluate the nite series.

evaluate the nite series.

evaluate the nite series.

, evaluate the innite series, if possible.

, evaluate the innite series, if possible.

, evaluate the innite series, if possible.

, evaluate the innite series, if possible.

, apply sigma notation to write the sum.

, apply sigma notation to write the sum.

, apply sigma notation to write the sum.

, apply sigma notation to write the sum.1 + 2 + 3 + 4 + 5 + + 21 + 22 + 231

, apply sigma notation to write the sum.

, apply sigma notation to write the sum.

, apply sigma notation to write the sum.

apply sigma notation to write the sum.x + x2 + x3 2 + x4 6 + x5 24 + x6 120 x + x2 + x3 2 + x4 6 + x5 24 + x6 120

Money. Upon graduation Jessica receives a commission from the U.S. Navy to become an ofcer and a $20,000 signing bonus for selecting aviation. She puts the entire bonus in an account that earns 6% interest compounded monthly. The balance in the account after n months is Her commitment to the Navy is 6 years. Calculate What does represent?

Money. Dylan sells his car in his freshman year and puts $7000 in an account that earns 5% interest compounded quarterly. The balance in the account after n quarters is Calculate What does represent?

Salary.An attorney is trying to calculate the costs associated with going into private practice. If she hires a paralegal to assist her, she will have to pay the paralegal $20 per hour. To be competitive with most rms, she will have to give her paralegal a $2 per hour raise per year. Find a general term of a sequence which represents the hourly salary of a paralegal with n years of experience. What will be the paralegals salary with 20 years of experience?

NFL Salaries.A player in the NFL typically has a career that lasts 3 years. The practice squad makes the league minimum of $275,000 (2004) in the rst year, with a $75,000 raise per year. Write the general term of a sequence that represents the salary of an NFL player making the league minimum during his entire career. Assuming corresponds to the rst year, what does represent?

Salary. Upon graduation Sheldon decides to go to work for a local police department. His starting salary is $30,000 per year, and he expects to get a 3% raise per year. Write the recursion formula for a sequence that represents his salary n years on the job. Assume n  0 represents his rst year making $30,000.

Escherichia coli.A single cell of bacteria reproduces through a process called binary ssion. Escherichia coli cells divide into two every 20 minutes. Suppose the same rate of division is maintained for 12 hours after the original cell enters the body. How many E. coli bacteria cells would be in the body 12 hours later? Suppose there is aninnitenutrientsourceso thattheE.colibacteriamaintainthesame rate of division for 48 hours after the original cell enters the body. How many E. coli bacteria cells would be in the body 48 hours later

AIDS/HIV.A typical person has 500 to 1500 T cells per drop of blood in the body. HIV destroys the T cell count at a rate of 50100 cells per drop of blood per year, depending on how aggressive it is in the body. Generally, the onset of AIDS occurs once the bodys T cell count drops below 200. Write a sequence that represents the total number of T cells in a person infected with HIV. Assume that before infection the person has a 1000 T cell count and the rate at which the infection spreads corresponds to a loss of 75 T cells per drop of blood per year. How much time will elapse until this person has full-blown AIDS?

Company Sales. Lowes reported total sales from 2003 through 2004 in the billions. The sequence represents the total sales in billions of dollars. Assuming corresponds to 2003, what were the reported sales in 2003 and 2004? What does represent?

Cost of Eating Out.A college student tries to save money by bringing a bag lunch instead of eating out. He will be able to save $100 per month. He puts the money into his savings account, which draws 1.2% interest and is compounded monthly. The balance in his account after n months of bagging his lunch is Calculate the rst four terms of this sequence. Calculate the amount after 3 years (36 months).

Cost of Acrylic Nails.A college student tries to save money by growing her own nails out and not spending $50 per month on acrylic lls. She will be able to save $50 per month. She puts the money into her savings account, which draws 1.2% interest and is compounded monthly. The balance in her account after n months of natural nails is Calculate the rst four terms of this sequence. Calculate the amount after 4 years (48 months).

Math and Engineering. The formula can be used to approximate the function Compute the rst ve terms of this formula to approximate Apply the result to nd and compare this result with the calculator value of 88. Home Prices. If the ination rate is per year and the average price of a home is $195,000, the average price of a home after n years is given by Find the average price of the home after 6 years. An = 195,000(1.035)n. 3.5% e2.

Home Prices. If the ination rate is per year and the average price of a home is $195,000, the average price of a home after n years is given by Find the average price of the home after 6 years.

Approximating Functions. Polynomials can be used to approximate transcendental functions such as ln(x) and which are found in advanced mathematics and engineering. For example, can be used to approximate where x is close to 1. Use the rst ve terms of the series to approximate Next, nd and compare with the value given by your calculator.

Future Value of an Annuity.The future value of an ordinary annuity is given by the formula FV = PMT[((1 + i)n - 1)/i], where PMT amount paid into the account at the end of each period, rate per period, and of compounding periods. If you invest $5000 at the end of each year for 5 years, you will have an accumulated value of FV as given in the above equation at the end of the nth year. Determine how much is in the account at the end of each year for the next 5 years if i = 0.06.

explain the mistake that is made.Simplify the ratio of factorials: Solution: Express 6! in factored form. Cancel the 3! in the numerator and denominator. Write out the factorials. Simplify. What mistake was made

explain the mistake that is made.Simplify the factorial expression: Solution: Express factorials in factored form. Cancel common terms. This is incorrect. What mistake was made?

explain the mistake that is made. Find the rst four terms of the sequence dened by Solution: Find the term. Find the term. Find the term. Find the term. The sequence is incorrect. What mistake was made

explain the mistake that is made.Evaluate the series Solution: Write out the sum. Simplify the sum. This is incorrect. What mistake was made?

determine whether each statement is true or false.

determine whether each statement is true or false.

determine whether each statement is true or false.

determine whether each statement is true or false.

Write the rst four terms of the sequence dened by the recursion formula:

Write the rst four terms of the sequence dened by the recursion formula: D Z 0a n = Dan-1a 1 = C

Fibonacci Sequence. An explicit formula for the nth term of the Fibonacci sequence is: Apply algebra (not your calculator) to nd the rst two terms of this sequence and verify that these are indeed the rst two terms of the Fibonacci sequence.

Let for and Find the rst ve terms of this sequence and make a generalization for the nth term.

The sequence dened by approaches the number e as n gets large. Use a graphing calculator to nd and keep increasing n until the terms in the sequence approach 2.7183.

The Fibonacci sequence is dened by and for The ratio is an an+1 an n 3.a n = an-2 + an-1 a2 = 1,a 1 = 1, a10,000,a 1000,a 100, an = a1 + 1 n b n Understand the difference between an arithmetic sequence and an arithmetic series.SKILLS OBJECTIVES Recognize an arithmetic sequence. Find the general nth term of an arithmetic sequence. Evaluate a nite arithmetic series. Use arithmetic sequences and series to model real-world problems. SECTION 12.2 Arithmetic Sequences The word arithmetic(with emphasis on the third syllable) often implies adding or subtracting of numbers. Arithmetic sequences are sequences whose terms are found by adding a constant to each previous term. The sequence 1, 3, 5, 7, 9, . . . is arithmetic because each successive term is found by adding 2 to the previous term. approximation of the golden ratio. The ratio approaches a constant (phi) as n gets large. Find the golden ratio using a graphing utility.

Use a graphing calculator SUMto sum . Compare it with your answer to Exercise 61.

Use a graphing calculator SUM to sum . Compare it with your answer to Exercise 62.

determine whether the sequence is arithmetic. If it is, nd the common difference

determine whether the sequence is arithmetic. If it is, nd the common difference

determine whether the sequence is arithmetic. If it is, nd the common difference

determine whether the sequence is arithmetic. If it is, nd the common difference

determine whether the sequence is arithmetic. If it is, nd the common difference

determine whether the sequence is arithmetic. If it is, nd the common difference

determine whether the sequence is arithmetic. If it is, nd the common difference

determine whether the sequence is arithmetic. If it is, nd the common difference

determine whether the sequence is arithmetic. If it is, nd the common difference

determine whether the sequence is arithmetic. If it is, nd the common difference

nd the rst four terms of the sequence. Determine whether the sequence is arithmetic, and if so, nd the common difference.

nd the rst four terms of the sequence. Determine whether the sequence is arithmetic, and if so, nd the common difference.

nd the rst four terms of the sequence. Determine whether the sequence is arithmetic, and if so, nd the common difference.

nd the rst four terms of the sequence. Determine whether the sequence is arithmetic, and if so, nd the common difference.

nd the rst four terms of the sequence. Determine whether the sequence is arithmetic, and if so, nd the common difference.

nd the rst four terms of the sequence. Determine whether the sequence is arithmetic, and if so, nd the common difference.

nd the rst four terms of the sequence. Determine whether the sequence is arithmetic, and if so, nd the common difference.

nd the rst four terms of the sequence. Determine whether the sequence is arithmetic, and if so, nd the common difference.

nd the rst four terms of the sequence. Determine whether the sequence is arithmetic, and if so, nd the common difference.

nd the rst four terms of the sequence. Determine whether the sequence is arithmetic, and if so, nd the common difference.

nd the specied term for each arithmetic sequence.

nd the specied term for each arithmetic sequence.

nd the specied term for each arithmetic sequence.

nd the specied term for each arithmetic sequence.

nd the specied term for each arithmetic sequence.

nd the specied term for each arithmetic sequence.

nd the specied term for each arithmetic sequence.

nd the specied term for each arithmetic sequence.

nd the specied term for each arithmetic sequence The 10th term of the sequence

nd the specied term for each arithmetic sequence 31. The 100th term of the sequence 33. The 21st term of the sequence 30. The 19th term of the sequence

nd the specied term for each arithmetic sequenceThe 100th term of the sequence

nd the specied term for each arithmetic sequence

nd the specied term for each arithmetic sequence

nd the specied term for each arithmetic sequence The 33rd term of the sequence 1 5, 8 15, 13 15, 6 5

for each arithmetic sequence described, nd a1 and d and construct the sequence by stating the general, or nth, term. The 5th term is 44 and the 17th term is 152.

for each arithmetic sequence described, nd a1 and d and construct the sequence by stating the general, or nth, term. The 9th term is and the 21st term is

for each arithmetic sequence described, nd a1 and d and construct the sequence by stating the general, or nth, term.The 7th term is and the 17th term is

for each arithmetic sequence described, nd a1 and d and construct the sequence by stating the general, or nth, term. The 8th term is 47 and the 21st term is 112.

for each arithmetic sequence described, nd a1 and d and construct the sequence by stating the general, or nth, term. The 4th term is 3 and the 22nd term is 15.

for each arithmetic sequence described, nd a1 and d and construct the sequence by stating the general, or nth, term. The 11th term is and the 31st term is -13.

nd the sum.

nd the sum.

nd the sum.

nd the sum.

nd the sum.

nd the sum.

nd the sum.2 + 7 + 12 + 17 + + 62

nd the sum.

nd the sum.4 + 7 + 10 + + 151

nd the sum.

nd the sum.

nd the sum.

nd the partial sum of the arithmetic series. The rst 18 terms of .

nd the partial sum of the arithmetic series. The rst 21 terms of .

nd the partial sum of the arithmetic series. The rst 43 terms of

nd the partial sum of the arithmetic series. The rst 37 terms of

nd the partial sum of the arithmetic series. The rst 18 terms of .

nd the partial sum of the arithmetic series.

Comparing Salaries. Colin and Camden are twin brothers graduating with B.S. degrees in biology. Colin takes a job at the San Diego Zoo making $28,000 for his rst year with a $1500 raise per year every year after that. Camden accepts a job at Florida Fish and Wildlife making $25,000 with a guaranteed $2000 raise per year. How much will each of the brothers have made in a total of 10 years?

Comparing Salaries. On graduating with a Ph.D. in optical sciences, Jasmine and Megan choose different career paths. Jasmine accepts a faculty position at the University of Arizona making $80,000 with a guaranteed $2000 raise every year. Megan takes a job with the Boeing Corporation making $90,000 with a guaranteed $5000 raise each year. Calculate how many total dollars each woman will have made after 15 years.

Theater Seating.You walk into the premiere of Brad Pitts new movie, and the theater is packed, with almost every seat lled. You want to estimate the number of people in the theater. You quickly count to nd that there are 22 seats in the front row, and there are 25 rows in the theater. Each row appears to have 1 more seat than the row in front of it. How many seats are in that theater?

Field of Tulips. Every spring the Skagit County Tulip Festival plants more than 100,000 bulbs. In honor of the Tri-Delta sorority that has sent 120 sisters from the University of Washington to volunteer for the festival, Skagit County has planted tulips in the shape of In each of the triangles there are 20 rows of tulips, each row having one less than the row before. How many tulips are planted in each delta if there is 1 tulip in the rst row?

Worlds Largest Champagne Fountain. From December 28 to 30, 1999, Luuk Broos, director of Maison Luuk-Chalet Fontain, constructed a 56-story champagne fountain at the Steigenberger Kurhaus Hotel, Scheveningen, Netherlands. The fountain consisted of 30,856 champagne glasses. Assuming there was one glass at the top and the number of glasses in each row forms an arithmetic sequence, how many were on the bottom row (story)? How many glasses less did each successive row (story) have? Assume each story is one row.

Stacking of Logs. If 25 logs are laid side by side on the ground, and 24 logs are placed on top of those, and 23 logs are placed on the 3rd row, and the pattern continues until there is a single log on the 25th row, how many logs are in the stack?

Falling Object.When a skydiver jumps out of an airplane, she falls approximately 16 feet in the 1st second, 48 feet during the 2nd second, 80 feet during the 3rd second, 112 feet during the 4th second, and 144 feet during the 5th second, and this pattern continues. If she deploys her parachute after 10 seconds have elapsed, how far will she have fallen during those 10 seconds?

Falling Object. If a penny is dropped out of a plane, it falls approximately 4.9 meters during the 1st second, 14.7 meters during the 2nd second, 24.5 meters during the 3rd second, and 34.3 meters during the 4th second. Assuming this pattern continues, how many meters will the penny have fallen after 10 seconds?

Grocery Store.A grocer has a triangular display of oranges in a window. There are 20 oranges in the bottom row, and the number of oranges decreases by one in each row above this row. How many oranges are in the display?

Salary. Suppose your salary is $45,000 and you receive a $1500 raise for each year you work for 35 years. a. How much will you earn during the 35th year? b. What is the total amount you earned over your 35-year career?

Theater Seating.At a theater, seats are arranged in a triangular pattern of rows with each succeeding row having one more seat than the previous row. You count the number of seats in the fourth row and determine that there are 26 seats. a. How many seats are in the rst row? b. Now, suppose there are 30 rows of seats. How many total seats are there in the theater?

Mathematics. Find the exact sum of 1 e + 3 e + 5 e + + 23 e

Find the general or nth term of the arithmetic sequence 3, 4, 5, 6, 7, . . . . Solution: The common difference of this sequence is 1. The rst term is 3. The general term is This is incorrect. What mistake was made?

Find the general or nth term of the arithmetic sequence 10, 8, 6, . . . . Solution: The common difference of this sequence is 2. The rst term is 10. The general term is This is incorrect. What mistake was made?

Find the sum Solution: The sum is given by where Identify the 1st and 10th terms. Substitute and into This is incorrect. What mistake was made?

Find the sum . Solution: This is an arithmetic sequence with common difference of 6. The general term is given by 87 is the 15th term of the series. The sum of the series is This is incorrect. What mistake was made?

determine whether each statement is true or false. An arithmetic sequence and a nite arithmetic series are the same.

determine whether each statement is true or false. The sum of all innite and nite arithmetic series can always be found.

An alternating sequence cannot be an arithmetic sequence.

The common difference of an arithmetic sequence is always positive.

Find the sum

Find the sum a30 k=-29 ln ek.

The wave number, v (reciprocal of wave length) of certain light waves in the spectrum of light emitted by hydrogen is given by where A series of lines is given by holding k constant and varying the value of n. Suppose and Find the limit of the wave number of the series.

In a certain arithmetic sequence and If nd the value of n.

Use a graphing calculator SUM to sum the natural numbers from 1 to 100.

Use a graphing calculator to sum the even natural numbers from 1 to 100.

Use a graphing calculator to sum the odd natural numbers from 1 to 100.

Use a graphing calculator to sum Compare it with your answer to Exercise 43.

Use a graphing calculator to sum

Use a graphing calculator to sum

determine whether the sequence is geometric. If it is, nd the common ratio.

determine whether the sequence is geometric. If it is, nd the common ratio.

determine whether the sequence is geometric. If it is, nd the common ratio.

determine whether the sequence is geometric. If it is, nd the common ratio.

determine whether the sequence is geometric. If it is, nd the common ratio.

determine whether the sequence is geometric. If it is, nd the common ratio.

determine whether the sequence is geometric. If it is, nd the common ratio.

determine whether the sequence is geometric. If it is, nd the common ratio.

write the rst ve terms of the geometric series.

write the rst ve terms of the geometric series.

write the rst ve terms of the geometric series.

write the rst ve terms of the geometric series.

write the rst ve terms of the geometric series.

write the rst ve terms of the geometric series.

write the rst ve terms of the geometric series.

write the rst ve terms of the geometric series.

write the formula for the nth term of the geometric series.

write the formula for the nth term of the geometric series.

write the formula for the nth term of the geometric series.

write the formula for the nth term of the geometric series.

write the formula for the nth term of the geometric series.

write the formula for the nth term of the geometric series.

write the formula for the nth term of the geometric series.

write the formula for the nth term of the geometric series.

nd the indicated term of the geometric sequence. 7th term of the sequence

nd the indicated term of the geometric sequence. 10th term of the sequence

nd the indicated term of the geometric sequence. 13th term of the sequence

nd the indicated term of the geometric sequence. 9th term of the sequence

nd the indicated term of the geometric sequence. 15th term of the sequence

nd the indicated term of the geometric sequence. 8th term of the sequence

nd the sum of the nite geometric series.

nd the sum of the nite geometric series.

nd the sum of the nite geometric series.

nd the sum of the nite geometric series.

nd the sum of the nite geometric series.

nd the sum of the nite geometric series.

nd the sum of the nite geometric series.

nd the sum of the nite geometric series.

nd the sum of the nite geometric series.

nd the sum of the nite geometric series.

nd the sum of the innite geometric series, if possible.

nd the sum of the innite geometric series, if possible.

nd the sum of the innite geometric series, if possible.

nd the sum of the innite geometric series, if possible.

nd the sum of the innite geometric series, if possible.

nd the sum of the innite geometric series, if possible.

nd the sum of the innite geometric series, if possible.

nd the sum of the innite geometric series, if possible.

nd the sum of the innite geometric series, if possible.

nd the sum of the innite geometric series, if possible.

nd the sum of the innite geometric series, if possible.

nd the sum of the innite geometric series, if possible.

nd the sum of the innite geometric series, if possible.

nd the sum of the innite geometric series, if possible.

Salary. Jeremy is offered a government job with the Department of Commerce. He is hired on the GS scale at a base rate of $34,000 with 2.5% increases in his salary per year. Calculate what his salary will be after he has been with the Department of Commerce for 12 years

Salary.Alison is offered a job with a small start-up company that wants to promote loyalty to the company with incentives for employees to stay with the company. The company offers her a starting salary of $22,000 with a guaranteed 15% raise per year. What will her salary be after she has been with the company for 10 years?

Depreciation. Brittany, a graduating senior in high school, receives a laptop computer as a graduation gift from herAunt Jeanine so that she can use it when she gets to the University of Alabama. If the laptop costs $2000 new and depreciates 50% per year, write a formula for the value of the laptop n years after it was purchased. How much will the laptop be worth when Brittany graduates from college (assuming she will graduate in 4 years)? How much will it be worth when she nishes graduate school? Assume graduate school is another 3 years.

Depreciation. Derek is deciding between a new Honda Accord and the BMW 325 series. The BMW costs $35,000 and the Honda costs $25,000. If the BMW depreciates at 20% per year and the Honda depreciates at 10% per year, nd formulas for the value of each car n years after it is purchased. Which car is worth more in 10 years?

Bungee Jumping.A bungee jumper rebounds 70% of the height jumped. Assuming the bungee jump is made with a cord that stretches to 100 feet, how far will the bungee jumper travel upward on the fth rebound?

Bungee Jumping.A bungee jumper rebounds 65% of the height jumped. Assuming the bungee cord stretches 200 feet, how far will the bungee jumper travel upward on the eighth rebound?

Population Growth. One of the fastest-growing universities in the country is the University of Central Florida. The student populations each year starting in 2000 were 36,000, 37,800, 39,690, 41,675, If this rate continued, how many students were at UCF in 2010?

Website Hits. The website for Matchbox 20 (www. matchboxtwenty.com) has noticed that every week the number of hits to its website increases 5%. If there were 20,000 hits this week, how many will there be exactly 52 weeks from now?

Rich Mans Promise.A rich man promises that he will give you $1000 on January 1, and every day after that, he will pay you 90% of what he paid you the day before. How many days will it take before you are making less than $1? How much will the rich man pay out for the entire month of January? Round answers to the nearest dollar.

Poor Mans Clever Deal.A poor man promises to work for you for $0.01 the rst day,$0.02 on the second day,$0.04 on the third day; his salary will continue to double each day. If he started on January 1,how much would he be paid to work on January 31? How much total would he make during the month? Round answers to the nearest dollar.

Investing Lunch.A newlywed couple decides to stop going out to lunch every day and instead brings their lunch. They estimate it will save them $100 per month. They invest that $100 on the rst of every month into an account that is compounded monthly and pays 5% interest. How much will be in the account at the end of 3 years?

Pizza as an Investment.A college freshman decides to stop ordering late-night pizzas (for both health and cost reasons). He realizes that he has been spending $50 a week on pizzas. Instead, he deposits $50 into an account that compounds weekly and pays 4% interest. How much money will be in the account in 52 weeks?

Tax-Deferred Annuity. Dr. Schober contributes $500 from her paycheck (weekly) to a tax-deferred investment account. Assuming the investment earns 6% and is compounded weekly, how much will be in the account in 26 weeks? 52 weeks?

Saving for a House. If a new graduate decides she wants to save for a house and she is able to put $300 every month into an account that earns 5% compounded monthly, how much will she have in the account in 5 years?

House Values. In 2008, you buy a house for $195,000. The value of the house appreciates 6.5% per year, on the average. How much is the house worth after 15 years?

The Bouncing Ball Problem.A ball is dropped from a height of 9 feet. Assume that on each bounce,the ball rebounds to one-third of its previous height. Find the total distance that the ball travels.

Probability. A fair coin is tossed repeatedly. The probability that the rst head occurs on the nth toss is given by the function where Show that

Salary. Suppose you work for a supervisor who gives you two different options to choose from for your monthly pay. Option 1: The company pays you $0.01 for the rst day of work, $0.02 the second day, $0.04 for the third day, $0.08 for the fourth day, and so on for 30 days. Option 2:You can receive a check right now for $10 million. Which pay option is better? How much better is it?

explain the mistake that is made.nd the nth term of the geometric sequence: Solution: Identify the rst term and common ratio. and Substitute and into Simplify. This is incorrect. What mistake was made?

explain the mistake that is made.Find the sum of the rst n terms of the nite geometric series: Solution: Write the sum in sigma notation. Identify the rst term and common ratio. and Substitute and into Simplify. This is incorrect. What mistake was made?

explain the mistake that is made.Find the sum of the nite geometric series Solution: Identify the rst term and common ratio. and Substitute and into Simplify. Substitute This is incorrect. What mistake was made?

explain the mistake that is made.Find the sum of the innite geometric series Solution: Identify the rst term and common ratio. and Substitute and into Simplify. This is incorrect. The series does not sum to What mistake was made?

determine whether each statement is true or false. An alternating sequence cannot be a geometric sequence.

determine whether each statement is true or false. All nite and innite geometric series can always be evaluated.

determine whether each statement is true or false. The common ratio of a geometric sequence can be positive or negative.

determine whether each statement is true or false. An innite geometric series can be evaluated if the common ratio is less than or equal to 1.

State the conditions for the sum to exist. Assuming those conditions are met, nd the sum

Find the sum of

Represent the repeating decimal 0.474747. . . as a fraction (ratio of two integers).

Suppose the sum of an innite geometric series is where x is a variable. a. Write out the rst ve terms of the series. b. For what values of x will the series converge?

Sum the series: Apply a graphing utility to conrm your answer.

Does the sum of the innite series exist? Use a graphing calculator to nd it.

Apply a graphing utility to plot and and let the range of x be Based on what you see, what do you expect the geometric series to sum to?

Apply a graphing utility to plot and and let x range from Based on what you see, what do you expect the geometric series to sum to?

Apply a graphing utility to plot and and let x range from Based on what you see, what do you expect the geometric series to sum to?

prove the statements using mathematical induction for all positive integers n.

prove the statements using mathematical induction for all positive integers n.

prove the statements using mathematical induction for all positive integers n.

prove the statements using mathematical induction for all positive integers n.

prove the statements using mathematical induction for all positive integers n.

prove the statements using mathematical induction for all positive integers n.

prove the statements using mathematical induction for all positive integers n. is divisible by 3.

prove the statements using mathematical induction for all positive integers n. is divisible by 3.

prove the statements using mathematical induction for all positive integers n. is divisible by 2.

prove the statements using mathematical induction for all positive integers n.

prove the statements using mathematical induction for all positive integers n.

prove the statements using mathematical induction for all positive integers n.

prove the statements using mathematical induction for all positive integers n.

prove the statements using mathematical induction for all positive integers n.

prove the statements using mathematical induction for all positive integers n.

prove the statements using mathematical induction for all positive integers n.

prove the statements using mathematical induction for all positive integers n.

prove the statements using mathematical induction for all positive integers n.

prove the statements using mathematical induction for all positive integers n.

prove the statements using mathematical induction for all positive integers n.

prove the statements using mathematical induction for all positive integers n.

prove the statements using mathematical induction for all positive integers n.

prove the statements using mathematical induction for all positive integers n. The sum of an arithmetic sequence:

prove the statements using mathematical induction for all positive integers n. The sum of a geometric sequence: a1 + a1r + a1r2 + + a1rn-1 = a1 a1 - rn 1 - r b.

The Tower of Hanoi. This is a game with three pegs and n disks (largest on the bottom and smallest on the top). The goal is to move this entire tower of disks to another peg (in the same order). The challenge is that you may move only one disk at a time, and at no time can a larger disk be resting on a smaller disk. You may want to rst go online to www.mazeworks.com/hanoi/index/htm and play the game.What is the smallest number of moves needed if there are three disks?

The Tower of Hanoi. This is a game with three pegs and n disks (largest on the bottom and smallest on the top). The goal is to move this entire tower of disks to another peg (in the same order). The challenge is that you may move only one disk at a time, and at no time can a larger disk be resting on a smaller disk. You may want to rst go online to www.mazeworks.com/hanoi/index/htm and play the game. What is the smallest number of moves needed if there are four disks?

The Tower of Hanoi. This is a game with three pegs and n disks (largest on the bottom and smallest on the top). The goal is to move this entire tower of disks to another peg (in the same order). The challenge is that you may move only one disk at a time, and at no time can a larger disk be resting on a smaller disk. You may want to rst go online to www.mazeworks.com/hanoi/index/htm and play the game.What is the smallest number of moves needed if there are ve disks?

The Tower of Hanoi. This is a game with three pegs and n disks (largest on the bottom and smallest on the top). The goal is to move this entire tower of disks to another peg (in the same order). The challenge is that you may move only one disk at a time, and at no time can a larger disk be resting on a smaller disk. You may want to rst go online to www.mazeworks.com/hanoi/index/htm and play the game.What is the smallest number of moves needed if there are n disks? Prove it by mathematical induction

Telephone Infrastructure. Suppose there are n cities that are to be connected with telephone wires. Apply mathematical induction to prove that the number of telephone wires required to connect the n cities is given by Assume each city has to connect directly with any other city. 30. Geometry. Prove, with mathematical induction, that the sum of the interior angles of a regular polygon of n sides is given by the formula: for Hint: Divide a polygon into triangles. For example, a four-sided polygon can be divided into two triangles. A ve-sided polygon can be divided into three triangles. A six-sided polygon can be divided into four triangles, and so on. n 3.( n - 2)(180) n(n - 1) 2

The Tower of Hanoi. This is a game with three pegs and n disks (largest on the bottom and smallest on the top). The goal is to move this entire tower of disks to another peg (in the same order). The challenge is that you may move only one disk at a time, and at no time can a larger disk be resting on a smaller disk. You may want to rst go online to www.mazeworks.com/hanoi/index/htm and play the game.Geometry. Prove, with mathematical induction, that the sum of the interior angles of a regular polygon of n sides is given by the formula: for Hint: Divide a polygon into triangles. For example, a four-sided polygon can be divided into two triangles. A ve-sided polygon can be divided into three triangles. A six-sided polygon can be divided into four triangles, and so on.

determine whether each statement is true or false. Assume is true. If it can be shown that is true, then is true for all n, where n is any positive integer.

determine whether each statement is true or false. Assume is true. If it can be shown that and are true, then is true for all n, where n is any positive integer.

Apply mathematical induction to prove:

Apply mathematical induction to prove: a n k=1 k5 = n2(n + 1)2A2n2 + 2n - 1B 12

Apply mathematical induction to prove:

Use a graphing calculator to sum the series (1 2)  (2 3)  (3 4)  . . .  n(n  1) on the left side,and

evaluate the expression on the right side for . Do they agree with each other? Do your answers conrm the proof for Exercise 19?

Use a graphing calculator to sum the series  . . .  on the left side, and evaluate the expression on the right side for . Do they agree with each other? Do your answers conrm the proof for Exercise 22?

evaluate the binomial coefcients.

evaluate the binomial coefcients.

evaluate the binomial coefcients.

evaluate the binomial coefcients.

evaluate the binomial coefcients.

evaluate the binomial coefcients.

evaluate the binomial coefcients.

evaluate the binomial coefcients.

evaluate the binomial coefcients.

evaluate the binomial coefcients.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using the binomial theorem.

expand the expression using Pascals triangle.

expand the expression using Pascals triangle.

expand the expression using Pascals triangle.

expand the expression using Pascals triangle.

nd the coefcient C of the term in the binomial expansion.

nd the coefcient C of the term in the binomial expansion.

nd the coefcient C of the term in the binomial expansion.

nd the coefcient C of the term in the binomial expansion.

nd the coefcient C of the term in the binomial expansion.

nd the coefcient C of the term in the binomial expansion.

nd the coefcient C of the term in the binomial expansion.

nd the coefcient C of the term in the binomial expansion.

you will learn the n choose k notation for combinations. Lottery. In a state lottery in which six numbers are drawn from a possible 40 numbers, the number of possible six-number combinations is equal to How many possible combinations are there?

you will learn the n choose k notation for combinations. Lottery. In a state lottery in which six numbers are drawn from a possible 60 numbers,the number of possible six-number combinations is equal to How many possible combinations are there?

you will learn the n choose k notation for combinations. Poker.With a deck of 52 cards, 5 cards are dealt in a game of poker. There are a total of different 5-card poker hands that can be dealt. How many possible hands are there?

you will learn the n choose k notation for combinations. Canasta. In the card game canasta, two decks of cards including the jokers are used and 11 cards are dealt to each person. There are a total of different 11-card canasta hands that can be dealt. How many possible hands are there?

explain the mistake that is made. Evaluate the expression Solution: Write out the binomial coefcient in terms of factorials. Write out the factorials. Simplify. This is incorrect. What mistake was made?

explain the mistake that is made. Expand Solution: Write out with blanks. (x  2y)4  x4  x3y  x2y2  xy3  y4 Write out the terms from the fth row of Pascals triangle. 1, 4, 6, 4, 1 Substitute these coefcients into the binomial expansion. This is incorrect. What mistake was made?

determine whether each statement is true or false. The binomial expansion of has 10 terms.

determine whether each statement is true or false. The binomial expansion of has 16 terms.

determine whether each statement is true or false.

determine whether each statement is true or false.

Show that if 0 k n.

Show that if n is a positive integer, then: Hint: Let and use the binomial theorem to expand.

With a graphing utility, plot and in the same viewing screen. What is the binomial expansion of

With a graphing utility, plot and What is the binomial expansion of

With a graphing utility, plot and for What do you notice happening each time an additional term is added? Now, let Does the same thing happen?

With a graphing utility, plot and for What do you notice happening each time an additional term is added? Now, let Does the same thing happen?

With a graphing utility, plot and for What do you notice happening each time an additional term is added? Now, let Does the same thing happen?

With a graphing utility, plot and for What do you notice happening each time an additional term is added? Now, let Does the same thing happen?

use the formula for to evaluate each expression.

use the formula for to evaluate each expression.

use the formula for to evaluate each expression.

use the formula for to evaluate each expression.

use the formula for to evaluate each expression.

use the formula for to evaluate each expression.

use the formula for to evaluate each expression.

use the formula for to evaluate each expression.

use the formula for to evaluate each expression.10C5

use the formula for to evaluate each expression.9C4

use the formula for to evaluate each expression.50C6

use the formula for to evaluate each expression.50C10

use the formula for to evaluate each expression.

use the formula for to evaluate each expression.. 8C8

use the formula for to evaluate each expression.30C4

use the formula for to evaluate each expression.13C5

use the formula for to evaluate each expression.45C8

use the formula for to evaluate each expression.30C4

Computers.At the www.dell.com website, a customer can build a system. If there are four monitors to choose from, three different computers, and two different keyboards, how many possible system congurations are there?

Houses. In a new home community, a person can select from one of four models, ve paint colors, three tile selections, and two landscaping options. How many different houses (interior and exterior) are there to choose from?

Wedding Invitations.An engaged couple is ordering wedding invitations. The wedding invitations come in white or ivory. The writing can be printed, embossed, or engraved. The envelopes can come with liners or without. How many possible designs of wedding invitations are there to choose from?

Dinner. Siblings are planning their fathers 65th birthday dinner and have to select one of four main courses (baked chicken, grilled mahi-mahi, beef Wellington, or lasagna), one of two starches (rosemary potatoes or rice), one of three vegetables (green beans, carrots, or zucchini), and one of ve appetizers (soup, salad, pot stickers, artichoke dip, or calamari). How many possible dinner combinations are there?

PIN Number. Most banks require a 4-digit ATM PIN code for each customers bank card. How many possible four-digit PIN codes are there to choose from?

Password.All e-mail accounts require passwords. If a four-character password is required that can contain letters (but no numbers), how many possible passwords can there be? (Assume letters are not case sensitive.)

Leadership. There are 15 professors in a department and there are four leadership positions (chair, assistant chair, undergraduate coordinator, and graduate coordinator). How many possible leadership teams are there?

Fraternity Elections.A fraternity is having elections. There are three men running for president, two men running for vice-president, four men running for secretary, and one man running for treasurer. How many possible outcomes do the elections have?

Multiple-Choice Tests. There are 20 questions on a multiple-choice exam, and each question has four possible answers (A, B, C, and D). Assuming no answers are left blank, how many different ways can you answer the questions on the exam?

Multiple-Choice Tests. There are 25 questions on a multiple-choice exam, and each question has ve possible answers (A, B, C, D, and E). Assuming no answers are left blank, how many different ways can you answer the questions on the exam?

Zip Codes. In the United States a 5-digit zip code is used to route mail. How many possible 5-digit zip codes are possible? (All numbers can be used.) If 0s were eliminated from the rst and last digits, how many possible zip codes would there be?

License Plates. In a particular state there are six characters in a license plate: three letters followed by three numbers. If 0s and 1s are eliminated from possible numbers and Os and Is are eliminated from possible letters, how many different license plates can be made?

Class Seating. If there are 30 students in a class and there are exactly 30 seats, how many possible seating charts can be made, assuming all 30 students are present?

Season Tickets. Four friends buy four season tickets to the Green Bay Packers. To be fair, they change the seating arrangement every game. How many different seating arrangements are there for the four friends?

Combination Permutation Lock.A combination lock on most lockers will open when the correct choice of three numbers (1 to 40) is selected and entered in the correct order. Therefore, a combination lock should really be called a permutation lock. How many possible permutations are there, assuming no numbers can be repeated?

Safe.A safe will open when the correct choice of three numbers (1 to 50) is selected and entered in the correct order. How many possible permutations are there,assuming no numbers can be repeated?

Rafe.A fundraiser rafe is held to benet cystic brosis research, and 1000 rafe tickets are sold. There are three prizes rafed off. First prize is a round-trip ticket on Delta Air Lines, second prize is a round of golf for four people at a local golf course, and third prize is a $50 gift certicate to Chilis. How many possible winning scenarios are there if all 1000 tickets are sold to different people

Ironman Triathlon. If 100 people compete in an ironman triathlon, how many possible placings are there (rst, second, and third place)?

Lotto. If a state lottery picks from 53 numbers and 6 numbers are selected, how many possible 6-number combinations are there?

Lotto. If a state lottery picks from 53 numbers and 5 numbers are selected, how many possible 5-number combinations are there?

Cards. In a deck of 52 cards, how many different 5-card hands can be dealt?

Cards. In a deck of 52 cards, how many different 7-card hands can be dealt?

Blackjack. In a single-deck blackjack game (52 cards), how many different 2-card combinations are there?

Blackjack. In a single deck, how many two-card combinations are there that equal 21: ace (worth 11) and a 10 or face cardjack, queen, or king?

March Madness. Every spring, the NCAA mens basketball tournament starts with 64 teams. After two rounds, it is down to the Sweet Sixteen, and after two more rounds, it is reduced to the Final Four. Once 64 teams are selected (but not yet put in brackets), how many possible scenarios are there for the Sweet Sixteen?

March Madness. Every spring, the NCAA mens basketball tournament starts with 64 teams. After two rounds, it is down to the Sweet Sixteen, and after two more rounds, it is reduced to the Final Four. Once the 64 teams are identied (but not yet put in brackets), how many possible scenarios are there for the Final Four?

NFL Playoffs. There are 32 teams in the National Football League (16 AFC and 16 NFC). How many possible combinations are there for the Superbowl? (Assume one team from the AFC plays one team from the NFC in the Superbowl.)

NFL Playoffs.After the regular season in the National Football League, 12 teams make the playoffs (6 from the AFC and 6 from the NFC). How many possible combinations are there for the Superbowl once the 6 teams in each conference are identied?

Survivor. On the television show Survivor, one person is voted off the island every week. When it is down to six contestants, how many possible voting combinations remain, if no one will vote themself off the island? Assume that the order (who votes for whom) makes a difference. How many total possible voting outcomes are there?

American Idol. On the television show American Idol,a young rising star is eliminated from the competition every week. The rst week, each of the 12 contestants sings one song. How many possible ways could the contestants be ordered 112? How many possible ways could 6 men and 6 women be ordered to alternate female and male contestants?

Dancing with the Stars. In the popular TV show Dancing with the Stars, 12 entertainers (6 men and 6 women) compete in a dancing contest. The rst night, the show decides to select 3 men and 3 women. How many ways can this be done?

Dancing with the Stars. See Exercise 49. How many ways can six male celebrities line up for a picture alongside six female celebrities?

explain the mistake that is made. "In a lottery that picks from 30 numbers, how many ve-number combinations are there? Solution: Let and Calculate Simplify. This is incorrect. What mistake was made? "

explain the mistake that is made. "A homeowners association has 12 members on the board of directors. How many ways can the board elect a president, vice-president, secretary, and treasurer? Solution: Let and Calculate Simplify. This is incorrect. What mistake was made? "

determine whether each statement is true or false. The number of permutations of n objects is always greater than the number of combinations of n objects if the objects are distinct.

determine whether each statement is true or false. The number of permutations of n objects is always greater than the number of combinations of n objects even when the objects are indistinguishabl

determine whether each statement is true or false. The number of four-letter permutations of the letters A, B, C, and D is equal to the number of four-letter permutations of ABBA.

determine whether each statement is true or false. The number of possible answers to a true/false question is a permutation problem.

What is the relationship between and

What is the relationship between and nPr-1?

Simplify the expression .

What is the relationship between and nCn-r?

Employ a graphing utility with a feature and compare it with answers to Exercises 18.

Employ a graphing utility with a feature and compare it with answers to Exercises 918.

Use a graphing calculator to evaluate: a. b. c. Are answers in (a) and (b) the same? d. Why? 4!(10C4) 10P4

Use a graphing calculator to evaluate: a. b. c. Are answers in (a) and (b) the same? d. Why? 5!(12C5) 12P5

nd the sample space for each experiment. The sum of two dice rolled simultaneously.

nd the sample space for each experiment. A coin tossed three times in a row.

nd the sample space for each experiment. The sex (boy or girl) of four children born to the same parents.

nd the sample space for each experiment. Tossing a coin and rolling a die.

nd the sample space for each experiment. Two balls selected from a container that has 3 red balls, 2 blue balls, and 1 white ball

nd the sample space for each experiment. The grade (freshman, sophomore, or junior) of two high school students who work at a local restaurant.

nd the probability for the experiment of tossing a coin three times. Getting all heads.

nd the probability for the experiment of tossing a coin three times. Getting exactly one heads.

nd the probability for the experiment of tossing a coin three times. Getting at least one heads.

nd the probability for the experiment of tossing a coin three times. Getting more than one heads.

nd the probability for the experiment of tossing two dice. The sum is 3.

nd the probability for the experiment of tossing two dice. The sum is odd.

nd the probability for the experiment of tossing two dice. The sum is even

nd the probability for the experiment of tossing two dice. The sum is prime.

nd the probability for the experiment of tossing two dice. The sum is more than 7.

nd the probability for the experiment of tossing two dice. The sum is less than 7.

nd the probability for the experiment of drawing a single card from a deck of 52 cards. Drawing a non-face card.

nd the probability for the experiment of drawing a single card from a deck of 52 cards. Drawing a black card.

nd the probability for the experiment of drawing a single card from a deck of 52 cards. Drawing a 2, 4, 6, or 8.

nd the probability for the experiment of drawing a single card from a deck of 52 cards. Drawing a 3, 5, 7, 9, or ace.

let and and nd the probability of the event. Probability of not occurring.

let and and nd the probability of the event. Probability of not occurring.

let and and nd the probability of the event. Probability of either or occurring if and are mutually exclusive.

let and and nd the probability of the event. Probability of either or occurring if and are not mutually exclusive and

let and and nd the probability of the event. Probability of both and occurring if and are mutually exclusive

let and and nd the probability of the event. Probability of both and occurring if and are independent.

Cards.A deck of 52 cards is dealt. a. How many possible combinations of four-card hands are there? b. What is the probability of having all spades? c. What is the probability of having four of a kind?

Blackjack.A deck of 52 cards is dealt for blackjack. a. How many possible combinations of two-card hands are there? b. What is the probability of having 21 points (ace with a 10 or face card)?

Cards.With a 52-card deck,what is the probability of drawing a 7 or an 8?

Cards.With a 52-card deck,what is the probability of drawing a red 7 or a black 8?

Cards. By drawing twice, what is the probability of drawing a 7 and then an 8?

Cards. By drawing twice, what is the probability of drawing a red 7 and then a black 8?

Children.What is the probability of having ve daughters in a row and no sons?

Children. What is the probability of having four sons in a row and no daughters?

Children. What is the probability that of ve children at least one is a boy? Note: .

Children. What is the probability that of six children at least one is a girl? Note: .

Roulette. In roulette, there are 38 numbered slots (136, 0, and 00). Eighteen are red, 18 are black, and the 0 and 00 are green. What is the probability of having 4 reds in a row?

Roulette. What is the probability of having 2 greens in a row on a roulette table?

Item Defectiveness. For a particular brand of DVD players, 10% of the ones on the market are defective. If a company has ordered 8 DVD players, what is the probability that none of the 8 DVD players is defective

Item Defectiveness. For a particular brand of generators 20% of the ones on the market are defective. If a company buys 10 generators, what is the probability that none of the 10 generators is defective?

Number Generator. A random-number generator (computer program that selects numbers in no particular order) is used to select two numbers between 1 and 10. What is the probability that both numbers are even?

Number Generator. A random-number generator is used to select two numbers between 1 and 15. What is the probability that both numbers are odd?

assume each deal is from a complete (shufed) single deck of cards. Blackjack. What is the probability of being dealt a blackjack (any ace and any face card) with a single deck?

assume each deal is from a complete (shufed) single deck of cards. Blackjack. What is the probability of being dealt two blackjacks in a row with a single deck?

Sports. With the salary cap in the NFL, it is said that on any given Sunday any team could beat any other team. If we assume every week a team has a 50% chance of winning, what is the probability that a team will go 160?

Sports. With the salary cap in the NFL, it is said that on any given Sunday any team could beat any other team. If we assume every week a team has a 50% chance of winning, what is the probability that a team will have at least 1 win?

Genetics. Suppose both parents have the brown/blue pair of eye-color genes, and each parent contributes one gene to the child. Suppose the brown eye-color gene is dominant so that if the child has at least one brown gene, the color will dominate and the eyes will be brown. a. List the possible outcomes (sample space). Assume each outcome is equally likely. b. What is the probability that the child will have the blue/blue pair of genes? c. What is the probability that the child will have brown eyes?

Genetics. Refer to Exercise 47. In this exercise, the father has a brown/brown pair of eye-color genes, while the mother has a brown/blue pair of eye-color genes. a. List the possible outcomes (sample space). Assume each outcome is equally likely. b. What is the probability that the child will have a blue/blue pair of genes? c. What is the probability that the child will have brown eyes?

Playing Cards. How many 5-card hands can be drawn from a 52-card deck (no jokers)?

Playing Cards. a. How many ways can you select 2 aces and 3 other cards (non-aces) from a standard deck of 52 cards (no jokers)? b. What is the probability that you draw a 5-card hand with 2 aces and 3 non-aces?

Playing Cards. Find the probability of drawing 5 clubs from a standard deck of 52 cards.

Poker. Find the probability of getting 2 ves and 3 kings when drawing 5 cards from a standard deck of 52 cards.

explain the mistake that is made. Calculate the probability of drawing a 2 or a spade from a deck of 52 cards. Solution: The probability of drawing a 2 from a deck of 52 cards is The probability of drawing a spade from a deck of 52 cards is The probability of drawing a 2 or a spade is This is incorrect. What mistake was made?

explain the mistake that is made. Calculate the probability of having two boys and one girl. Solution: The probability of having a boy is The probability of having a girl is These are independent, so the probability of having two boys and a girl is This is incorrect. What mistake was made?

determine whether each statement is true or false. If and then must equal 0.1 if there are three possible events and they are all mutually exclusive.

determine whether each statement is true or false.If two events are mutually exclusive, then they cannot be independent.

determine whether each statement is true or false.If two events are independent, then they are not mutually exclusive.

determine whether each statement is true or false.The probability of having ve sons and no daughters is 1 minus the probability of having ve daughters and no sons.

If two people are selected at random, what is the probability that they have the same birthday? Assume 365 days per year.

If 30 people are selected at random, what is the probability that at least two of them will have the same birthday?

If one die is weighted so that 3 and 4 are the only numbers that the die will roll, and the other die is fair, what is the probability of rolling two dice that sum to 2, 5, or 6?

If one die is weighted so that 3 and 4 are the only numbers that the dice will roll, and 3 comes up twice as often as 4, what is the probability of rolling a 3?

Use a random-number generator on a graphing utility to select two numbers between 1 and 10. Run this generator 50 times. How many times (out of 50 trials) were both of the two numbers even? Compare with your answer from Exercise 41.

Use a random-number generator on a graphing utility to select two numbers between 1 and 15. Run this 50 times. How many times (out of 50 trials) were both of the two numbers odd? Compare with your answer from Exercise 42.

when a die is rolled once,the probability of getting a 2 is and the probability of not getting a 2 is . If a die is rolled ntimes,the probability of getting a 2 exactly ktimes can be found by using the binomial theorem:If a die is rolled 10 times, nd the probability of getting a 2 exactly two times. Round your answer to four decimal places.

when a die is rolled once,the probability of getting a 2 is and the probability of not getting a 2 is . If a die is rolled ntimes,the probability of getting a 2 exactly ktimes can be found by using the binomial theorem: If a die is rolled 8 times, nd the probability of getting a 2 at most two times. Round your answer to four decimal places.

Write the rst four terms of the sequence. Assume n starts at 1.

Write the rst four terms of the sequence. Assume n starts at 1.

Write the rst four terms of the sequence. Assume n starts at 1.

Write the rst four terms of the sequence. Assume n starts at 1.

Find the indicated term of the sequence.

Find the indicated term of the sequence.

Find the indicated term of the sequence.

Find the indicated term of the sequence.

Write an expression for the nth term of the given sequence.

Write an expression for the nth term of the given sequence. 1, 1 2, 3, 1 4, 5, 1 6, 7, 1 8,

Write an expression for the nth term of the given sequence.

Write an expression for the nth term of the given sequence. 1, 10, 102, 103,

Simplify the ratio of factorials.

Simplify the ratio of factorials.

Simplify the ratio of factorials.

Simplify the ratio of factorials.

Write the rst four terms of the sequence dened by the recursion formula.

Write the rst four terms of the sequence dened by the recursion formula.

Write the rst four terms of the sequence dened by the recursion formula.

Write the rst four terms of the sequence dened by the recursion formula.

Evaluate the nite series.

Evaluate the nite series.

Evaluate the nite series.

Evaluate the nite series.

Use sigma (summation) notation to write the sum.

Use sigma (summation) notation to write the sum. 2 + 4 + 6 + 8 + 10 + + 20

Use sigma (summation) notation to write the sum. 1 + x + x2 2 + x3 6 + x4 24 +

Use sigma (summation) notation to write the sum. x - x2 + x3 2 x4 6 + x5 24 x6 120 +

Marines Investment. With the prospect of continued ghting in Iraq, in December 2004, the Marine Corps offered bonuses of as much as $30,000in some cases, tax-freeto persuade enlisted personnel with combat experience and training to reenlist. Suppose a Marine put her entire $30,000 reenlistment bonus in an account that earns 4% interest compounded monthly. The balance in the account after n months is Her commitment with the Marines is 5 years. Calculate What does represent?

Sports. The NFL minimum salary for a rookie is $180,000. Suppose a rookie comes into the league making the minimum and gets a $30,000 raise every year he plays. Write the general term of a sequence that represents the salary of an NFL player making the league minimum during his entire career. Assuming corresponds to the rst year, what does represent?

Determine whether the sequence is arithmetic. If it is, nd the common difference.

Determine whether the sequence is arithmetic. If it is, nd the common difference. 13 + 23 + 33 +

Determine whether the sequence is arithmetic. If it is, nd the common difference.

Determine whether the sequence is arithmetic. If it is, nd the common difference. an =n +

Determine whether the sequence is arithmetic. If it is, nd the common difference.

Determine whether the sequence is arithmetic. If it is, nd the common difference.

Find the general, or nth, term of the arithmetic sequence given the rst term and the common difference.

Find the general, or nth, term of the arithmetic sequence given the rst term and the common difference.

Find the general, or nth, term of the arithmetic sequence given the rst term and the common difference.

Find the general, or nth, term of the arithmetic sequence given the rst term and the common difference.

For each arithmetic sequence described below, nd and d and construct the sequence by stating the general, or nth, term. The 5th term is 13 and the 17th term is 37.

For each arithmetic sequence described below, nd and d and construct the sequence by stating the general, or nth, term. The 7th term is and the 10th term is

For each arithmetic sequence described below, nd and d and construct the sequence by stating the general, or nth, term.The 8th term is 52 and the 21st term is 130.

For each arithmetic sequence described below, nd and d and construct the sequence by stating the general, or nth, term. The 11th term is and the 21st term is

Find the sum.

Find the sum.

Find the sum.2 + 8 + 14 + 20 + + 68

Find the sum.1 4 - 1 4 - 3 4 - - 31

Salary. Upon graduating with M.B.A.s, Bob and Tania opt for different career paths. Bob accepts a job with the U.S. Department of Transportation making $45,000 with a guaranteed $2000 raise every year. Tania takes a job with Templeton Corporation making $38,000 with a guaranteed $4000 raise every year. Calculate how many total dollars both Bob and Tania will have each made after 15 years.

Gravity.When a skydiver jumps out of an airplane, she falls approximately 16 feet in the 1st second, 48 feet during the 2nd second, 80 feet during the 3rd second, 112 feet during the 4th second, 144 feet during the 5th second, and this pattern continues. If she deploys her parachute after 5 seconds have elapsed, how far will she have fallen during those 5 seconds?

Determine whether the sequence is geometric. If it is, nd the common ratio.

Determine whether the sequence is geometric. If it is, nd the common ratio.

Determine whether the sequence is geometric. If it is, nd the common ratio.

Determine whether the sequence is geometric. If it is, nd the common ratio.

Write the rst ve terms of the geometric series.

Write the rst ve terms of the geometric series.

Write the rst ve terms of the geometric series.

Write the rst ve terms of the geometric series.

Write the formula for the nth term of the geometric series.

Write the formula for the nth term of the geometric series.

Write the formula for the nth term of the geometric series.

Write the formula for the nth term of the geometric series.

Find the indicated term of the geometric sequence. 25th term of the sequence

Find the indicated term of the geometric sequence. 10th term of the sequence

Find the indicated term of the geometric sequence. 12th term of the sequence

Find the indicated term of the geometric sequence. 11th term of the sequence

Evaluate the geometric series, if possible.

Evaluate the geometric series, if possible.

Evaluate the geometric series, if possible.

Evaluate the geometric series, if possible.

Evaluate the geometric series, if possible.

Evaluate the geometric series, if possible.

Salary. Murad is uent in four languages and is offered a job with the U.S. government as a translator. He is hired on the GS scale at a base rate of $48,000 with 2% increases in his salary per year. Calculate what his salary will be after he has been with the U.S. government for 12 years.

Boat Depreciation. Upon graduating from Auburn University, Philip and Steve get jobs at Disney Ride and Show Engineering and decide to buy a ski boat together. If the boat costs $15,000 new, and depreciates 20% per year, write a formula for the value of the boat n years after it was purchased. How much will the boat be worth when Philip and Steve have been working at Disney for 3 years?

Prove the statements using mathematical induction for all positive integers n.

Prove the statements using mathematical induction for all positive integers n.

Prove the statements using mathematical induction for all positive integers n. 2 + 7 + 12 + 17 + + (5n - 3) = n 2 (5n - 1)

Prove the statements using mathematical induction for all positive integers n. n 32 n2 7 (n + 1)2

Evaluate the binomial coefcients.

Evaluate the binomial coefcients.

Evaluate the binomial coefcients.

Evaluate the binomial coefcients.

Expand the expression using the binomial theorem.

Expand the expression using the binomial theorem.

Expand the expression using the binomial theorem.

Expand the expression using the binomial theorem.

Expand the expression using the binomial theorem.

Expand the expression using the binomial theorem.

Expand the expression using Pascals triangle.

Expand the expression using Pascals triangle.

Find the coefcient C of the term in the binomial expansion.

Find the coefcient C of the term in the binomial expansion.

Find the coefcient C of the term in the binomial expansion.

Find the coefcient C of the term in the binomial expansion.

Lottery. In a state lottery in which 6 numbers are drawn from a possible 53 numbers, the number of possible 6-number combinations is equal to How many possible combinations are there?

Canasta. In the card game canasta, two decks of cards including the jokers are used, and 13 cards are dealt to each person. A total of different 13-card canasta hands can be dealt. How many possible hands are there?

Use the formula for to evaluate each expression.

Use the formula for to evaluate each expression.

Use the formula for to evaluate each expression.

Use the formula for to evaluate each expression.

Use the formula for to evaluate each expression.

Use the formula for to evaluate each expression.

Use the formula for to evaluate each expression.

Use the formula for to evaluate each expression.

Car Options. A new Honda Accord comes in three models (LX,VX, and EX). Each of those models comes with either a cloth or a leather interior, and the exterior comes in either silver, white, black, red, or blue. How many different cars (models, interior seat upholstery, and exterior color) are there to choose from?

E-mail Passwords. All e-mail accounts require passwords. If a six-character password is required that can contain letters (but no numbers), how many possible passwords can there be if letters can be repeated? (Assume no letters are case-sensitive.)

Team Arrangements. There are 10 candidates for the board of directors, and there are four leadership positions (president, vice president, secretary, and treasurer). How many possible leadership teams are there?

License Plates. In a particular state, there are six characters in a license plate consisting of letters and numbers. If 0s and 1s are eliminated from possible numbers and Os and Is are eliminated from possible letters, how many different license plates can be made?

Seating Arrangements. Five friends buy ve season tickets to the Philadelphia Eagles. To be fair, they change the seating arrangement every game. How many different seating arrangements are there for the ve friends? How many seasons would they have to buy tickets in order to sit in all of the combinations (each season has eight home games)

Safe. A safe will open when the correct choice of three numbers (1 to 60) is selected in a specic order. How many possible permutations are there?

Rafe. A fundraiser rafe is held to benet the Make a Wish Foundation, and 100 rafe tickets are sold. Four prizes are rafed off. First prize is a round-trip ticket on American Airlines, second prize is a round of golf for four people at a Links golf course, third prize is a $100 gift certicate to the Outback Steakhouse, and fourth prize is a half-hour massage. How many possible winning scenarios are there if all 100 tickets are sold to different people?

Sports. There are 117 Division 1-A football teams in the United States. At the end of the regular season is the Bowl Championship Series, and the top two teams play each other in the championship game. Assuming that any two Division 1-A teams can advance to the championship, how many possible matchups are there for the championship game?

Cards. In a deck of 52 cards, how many different 6-card hands can be dealt?

Blackjack. In a game of two-deck blackjack (104 cards), how many 2-card combinations are there that equal 21, that is, ace and a 10 or face cardjack, queen, or king?

Coin Tossing. For the experiment of tossing a coin four times, what is the probability of getting all heads?

Dice. For an experiment of tossing two dice, what is the probability that the sum of the dice is odd?

Dice. For an experiment of tossing two dice, what is the probability of not rolling a combined 7?

Cards. For a deck of 52 cards, what is the probability of drawing a diamond?

let and and nd the probability of the event. Probability. Find the probability of an event not occurring.

let and and nd the probability of the event.Probability. Find the probability of either or occurring if and are mutually exclusive.

let and and nd the probability of the event.Probability. Find the probability of either or occurring if and are not mutually exclusive and

let and and nd the probability of the event. Probability. Find the probability of both and occurring if and are independent.

Cards. With a 52-card deck, what is the probability of drawing an ace or a 2?

Cards. By drawing twice, what is the probability of drawing an ace and then a 2? (Assume that after the rst card is drawn it is not put back into the deck.)

Children. What is the probability that in a family of ve children at least one is a girl?

Sports. With the salary cap in the NFL, it is said that on any given Sunday any team could beat any other team. If we assume every week a team has a 50% chance of winning, what is the probability that a team will go 111?

Use a graphing calculator SUM to nd the sum of the series

Use a graphing calculator SUM to nd the sum of the innite series if possible.

Use a graphing calculator to sum

Use a graphing calculator to sum

Apply a graphing utility to plot and and let x range from . Based on what you see, what do you expect the geometric series to sum to in this range of x values?

Does the sum of the innite series exist? Use a graphing calculator to nd it and round to four decimal places.

Use a graphing calculator to sum the series on the left side, and evaluate the expression on the right side for Do they agree with each other? Do your answers conrm the proof for Exercise 77?

Use a graphing calculator to plot the graphs of and in the by viewing rectangle. Do your answers conrm the proof for Exercise 78?

With a graphing utility, plot y1 1 8x, y2 1 8x 24x2, y3 1 8x 24x2 32x3, y4 1 8x 24x2 32x3 16x4,and for What do you notice happening each time an additional term is added? Now, let 0.1 x 1. Does the same thing happen?

With a graphing utility,plot and for What do you notice happening each time an additional term is added? Now, let 0.1 x 1. Does the same thing happen?

Use a graphing calculator to evaluate: a. b. 16C7 c. Are answers in (a) and (b) the same? d. Why?

Use a graphing calculator to evaluate: a. b. 52C6 c. Are answers in (a) and (b) the same? d. Why?

when two dice are rolled, the probability of getting a sum of 9 is and the probability of not getting a sum of 9 is . If two dice are rolled n times, the probability of getting a sum of 9 exactly k times can be found by using the binomial theorem . If two dice are rolled 10 times, nd the probability of getting a sum of 9 exactly three times. Round your answer to four decimal places. If a die is rolled 8 times,nd the probability of getting a sum of 9 at least two times. Round your answer to four decimal places.

when two dice are rolled, the probability of getting a sum of 9 is and the probability of not getting a sum of 9 is . If two dice are rolled n times, the probability of getting a sum of 9 exactly k times can be found by using the binomial theorem .

use the sequence 1, x, x2, x3, Write the nth term of the sequence.

use the sequence 1, x, x2, x3, Classify this sequence as arithmetic, geometric, or neither.

use the sequence 1, x, x2, x3, Find the nth partial sum of the series

use the sequence 1, x, x2, x3, Assuming this sequence is innite, write the series using sigma notation.

use the sequence 1, x, x2, x3, Assuming this sequence is innite, what condition would have to be satised in order for the sum to exist?

Find the following sum: .

Find the following sum: .

Find the following sum: .

Write the series using sigma notation, then nd its sum:2 + 7 + 12 + 17 + + 497.

Use mathematical induction to prove that 2 + 4 + 6 + + 2n = n2 + n.

Evaluate

Find the third term of

evaluate the expressions.

evaluate the expressions.

evaluate the expressions.

evaluate the expressions.

Expand the expression

Use the binomial theorem to expand the binomial

Explain why there are always more permutations than combinations.

What is the probability of not winning a trifecta (selecting the rst-, second-, and third-place nishers) in a horse race with 15 horses?

refer to a roulette wheel with 18 red, 18 black, and 2 green slots. Roulette. What is the probability of the ball landing in a redslot?

refer to a roulette wheel with 18 red, 18 black, and 2 green slots.Roulette. What is the probability of the ball landing in a red slot 5 times in a row?

refer to a roulette wheel with 18 red, 18 black, and 2 green slots.Roulette. If the four previous rolls landed on red, what is the probability that the next roll will land on red?

Marbles. If there are four red marbles,three blue marbles,two green marbles,and one black marble in a sack,nd the probability of pulling out the following order:black,blue,red,red,green.

Cards. What is the probability of drawing an ace or a diamond from a deck of 52 cards?

Human Anatomy. Vasopressin is a relatively simple protein that is found in the human liver. It consists of eight amino acids that must be joined together in one particular order for the effective functioning of the protein. a. How many different arrangements of the eight amino acids are possible? b. What is the probability of randomly selecting one of these arrangements and obtaining the correct arrangement to make vasopressin?

Find the constant term in the expression

Use a graphing calculator to sum a125 n=1C-11 4 + 5 6(n - 1)D.

Simplify

The length of a rectangle is 5 less than twice the width and the perimeter is 38 inches. What are the dimensions of the rectangle?

Solve using the quadratic formula:

Solve and express the solution in interval notation: .

Write an equation of the line with slope undened and x-intercept

Find the x-intercept and y-intercept and slope of the line

Using the function evaluate the difference quotient

Find the composite function and state the domain for and

Find the vertex of the parabola

Factor the polynomial as a product of linear factors.

Find the vertical and horizontal asymptotes of the function:

How much money should be put in a savings account now that earns a year compounded weekly, if you want to have $65,000 in 17 years?

Evaluate using the change-of-base formula. Round your answer to three decimal places.

Solve Round your answer to three decimal places.

Solve the system of linear equations. y = 8 5 x + 10 8x - 5y = 15

Solve the system of linear equations. 2x y z 1 x y 4z 3

Maximize the objective function subject to the constraints x + y 5, x 1, y 2. z = 4x + 5y

Solve the system using GaussJordan elimination.x + 5y - 2z = 3 x + y + 2z =3 2x - 4y + 4z = 10

Given nd C(A B).

Calculate the determinant.

Find the equation of a parabola with vertex (3, 5) and directrix

Graph x2 + y2 6 4.

Find the sum of the nite series

Classify the sequence as arithmetic, geometric, or neither: 5, 15, 45, 135, . .

There are 10 true/false questions on a quiz. Assuming no answers are left blank, how many different ways can you answer the questions on the quiz?

Use a graphing calculator to sum

Find the constant term in the expression .