Chapter 6

A ball is thrown upward so that it reaches a height of 9 feet and then falls to the ground. When it hits the ground, it bounces to \(\frac{1}{3}\) of its previous height. If the ball continues in this way, bouncing each time to \(\frac{1}{3}\) of its previous height until it comes to rest when it hits the ground for the fifth time, find the total distance the ball has traveled, starting from its highest point.

Show that \(\sum_{i=1}^{n} k a_{i}=k \sum_{i=1}^{n} a_{i}\)

What is the 10 th term of the geometric sequence \(0.25,0.5,1, \ldots ?\)

In \(19-30 :\) a. Write an algebraic expression that represents \(a_{n}\) for each sequence. b. Find the ninth term of each sequence. $$ \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \dots $$

Sarah wants to save for a special dress for the prom. The first month she saved \(\$ 15\) and each of the next five months she increased the amount that she saved by \(\$ 2 .\) What is the total amount Sarah saved over the six months?

If you start a job for which you are paid \(\$ 1\) the first day, \(\$ 2\) the second day, \(, \$ 4\) the third day, and so on, how many days will it take you to become a millionaire?

What is the 9 th term of the geometric sequence \(125,25,5, \ldots ?\)

In a theater, there are 20 seats in the first row. Each row has 3 more seats than the row ahead of it. There are 35 rows in the theater. Find the total number of seats in the theater.

In a theater, there are 20 seats in the first row. Each row has 3 more seats than the row ahead of it. There are 35 rows in the theater. a. Express the number of seats in the \(n\) th row of the theater in terms of \(n .\) b. Use sigma notation to represent the number of seats in the theater.

In a geometric sequence, \(a_{1}=1\) and \(a_{5}=16 .\) Find \(a_{9}\)

On Monday, Enid spent 45 minutes doing homework. On the remaining four days of the school week spent 15 minutes longer doing homework than she had the day before. Find the total number of minutes Enid spent doing homework from Monday to Friday.

On Monday, Elaine spent 45 minutes doing homework. On the remaining four days of the school week, she spent 15 minutes longer doing homework than she had the day before. a. Express the number of minutes Elaine spent doing homework on the \(n\) th day of the school week. b. Use sigma notation to represent the total number of minutes Elaine spent doing homework from Monday to Friday.

The first term of a geometric sequence is 1 and the 4 th term is \(27 .\) What is the 8 th term?

Keegan started a job that paid \(\$ 20,000\) a year. Each year after the first, he received a raise of \(\$ 600 .\) What was the total amount that Keegan earned in six years?

Use the graphing calculator to evaluate the following series to the nearest hundredth: $$ \begin{array}{llll}{\text { (1) } \sum_{n=1}^{50}\left(1+\frac{1}{n}\right)} & {\text { (2) } \sum_{k=1}^{18} \frac{5}{1+k}} & {\text { (3) } \sum_{n=1}^{20} \frac{(n-1)(-1)^{n}}{n}}\end{array} $$

In a geometric sequence, \(a_{1}=2\) and \(a_{3}=16 .\) Find \(a_{6}\)

In \(31-39,\) write the first five terms of each sequence. $$ a_{1}=5, a_{n}=a_{n-1}+1 $$

A new health food store's net income was a loss of \(\$ 2,300\) in its first month, but its net income increased by \(\$ 575\) in each succeeding month for the next year. What is the store's net income for the year?

In a geometric sequence, \(a_{3}=1\) and \(a_{7}=9 .\) Find \(a_{1}\)

In \(31-39,\) write the first five terms of each sequence. $$ a_{1}=1, a_{n+1}=3 a_{n} $$

Find two geometric means between 6 and \(93.75 .\)

In \(31-39,\) write the first five terms of each sequence. $$ a_{1}=1, a_{n}=2 a_{n-1}+1 $$

Find three geometric means between 3 and 9\(\frac{13}{27}\) .

In \(31-39,\) write the first five terms of each sequence. $$ a_{1}=-2, a_{n}=-2 a_{n-1} $$

Find three geometric means between 8 and \(2,592\)

In \(31-39,\) write the first five terms of each sequence. $$ a_{1}=20, a_{n}=a_{n-1}-4 $$

If \(\$ 1,000\) was invested at 6\(\%\) annual interest at the beginning of 2001 , list the geometric sequence that is the value of the investment at the beginning of each year from 2001 to \(2010 .\)

In \(31-39,\) write the first five terms of each sequence. $$ a_{1}=4, a_{n+1}=a_{n}+n $$

Al invested \(\$ 3,000\) in a certificate of deposit that pays 5\(\%\) interest per year. What is the value of the investment at the end of each of the first four years?

In \(31-39,\) write the first five terms of each sequence. $$ a_{2}=36, a_{n}=\frac{1}{3} a_{n-1} $$

In a small town, a census is taken at the beginning of each year. The census showed that there were \(5,000\) people living in the town at the beginning of 2001 and that the population decreased by 2\(\%\) each year for the next seven years. List the geometric sequence that gives the population of the town from 2001 to \(2008 .\) (A decrease of 2\(\%\) means that the population changed each year by a factor of \(0.98 .\) ) Write your answer to the nearest integer.

It is estimated that the deer population in a park was increasing by 10\(\%\) each year. If there were 50 deer in the park at the end of the first year in which a study was made, what is the estimated deer population for each of the next five years? Write your answer to the nearest integer.

In \(31-39,\) write the first five terms of each sequence. $$ a_{5}=\frac{1}{2}, a_{n}=\frac{1}{a_{n-1}} $$

A car that cost \(\$ 20,000\) depreciated by 20\(\%\) each year. Find the value of the car at the end of each of the first four years. (A depreciation of 20\(\%\) means that the value of the car each year was 0.80 times the value the previous year.)

Sean has started an exercise program. The first day he worked out for 30 minutes. Each day for the next six days, he increased his time by 5 minutes. a. Write the sequence for the number of minutes that Sean worked out for each of the seven days. b. Write a recursive definition for this sequence.

Sherri wants to increase her vocabulary. On Monday she learned the meanings of four new words. Each other day that week, she increased the number of new words that she learned by two. a. Write the sequence for the number of new words that Sherri learned each day for a week. b. Write a recursive definition for this sequence.

A manufacturing company purchases a machine for \(\$ 50,000\) . Each year the company estimates the depreciation to be 15\(\% .\) What will be the estimated value of the machine after each of the first six years?

Julie is trying to lose weight. She now weighs 180 pounds. Every week for eight weeks, she was able to lose 2 pounds. a. List Julie's weight for each week. b. Write a recursive definition for this sequence.

January \(1,2008,\) was a Tuesday. a. List the dates for each Tuesday in January of that year. b. Write a recursive definition for this sequence.

Hui started a new job with a weekly salary of \(\$ 400 .\) After one year, and for each year that followed, his salary was increased by 10\(\% .\) Hui left this job after six years. a. List the weekly salary that Hui earned each year. b. Write a recursive definition for this sequence.

One of the most famous sequences is the Fibonacci sequence. In this sequence, \(a_{1}=1, a_{2}=1,\) and for \(n>2, a_{n}=a_{n-2}+a_{n-1} .\) Write the first ten terms of this sequence.