Chapter 9

The sum of an infin te geometric series is five times the value of the fi st term. What is the common ratio of the series?

At which term does the sequence \(\\{10,12,14.4,17.28, \ldots\\}\) exceed \(100 ?\)

For the following exercises, find the number of terms in the given finite arithmetic sequence. $$ a=\left\\{\frac{1}{2}, 2, \frac{7}{2}, \ldots, 8\right\\} $$

The sum of an infinite geometric series is five times the value of the first term. What is the common ratio of the series?

Use this data for the exercises that follow: In 2013 , there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over). \(^{[34]}\) If you meet a U.S. citizen, what is the percent chance that the person is elderly? (Round to the nearest tenth of a percent.)

To get the best loan rates available, the Riches want to save enough money to place \(20 \%\) down on a \(\$ 160,000\) home. They plan to make monthly deposits of \(\$ 125\) in an investment account that offers \(8.5 \%\) annual interest compounded semiannually. Will the Riches have enough for a \(20 \%\) down payment after five years of saving? How much money will they have saved?

At which term does the sequence \(\left\\{\frac{1}{2187}, \frac{1}{729}, \frac{1}{243}, \frac{1}{81} \ldots\right\\}\) begin to have integer values?

Use this data for the exercises that follow: In 2013 , there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over). \(^{[34]}\) If you meet five U.S. citizens, what is the percent chance that exactly one is elderly? (Round to the nearest tenth of a percent.)

Karl has two years to save \(\$ 10,000\) to buy a used car when he graduates. To the nearest dollar, what would his monthly deposits need to be if he invests in an account offering a \(4.2 \%\) annual interest rate that compounds monthly?

For which term does the geometric sequence \(a_{n}=-36\left(\frac{2}{3}\right)^{n-1}\) fi st have a non-integer value?

For which term does the gcometric sequence \(a_{n}=-36\left(\frac{2}{3}\right)^{n-1}\) first have a non-integer value?

Follow these steps to evaluate a sequence defined recursively using a graphing calculator: • On the home screen, key in the value for the initial term \(a_{1}\) and press [ENTER]. • Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes \([2 \mathrm{ND}]\) ANS for the previous term \(a_{n-1}.\) Press \([\mathrm{ENTER}].\) • Continue pressing \([\text { ENTER }]\) to calculate the values for each successive term. Use the steps above to find the indicated term or terms for the sequence. Find the first five terms of the sequence \(a_{1}=\frac{87}{111},\) \(a_{n}=\frac{4}{3} a_{n-1}+\frac{12}{37} .\) Use the \(>\) Frac feature to give fractional results.

Use this data for the exercises that follow: In 2013 , there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over). \(^{[34]}\) If you meet five U.S. citizens, what is the percent chance that three are elderly? (Round to the nearest tenth of a percent.)

Keisha devised a week-long study plan to prepare for fi als. On the fi st day, she plans to study for 1 hour, and each successive day she will increase her study time by 30 minutes. How many hours will Keisha have studied after one week?

For the following exercises, use the information provided to graph the first 5 terms of the arithmetic sequence. $$ a_{1}=0, d=4 $$

Follow these steps to evaluate a sequence defined recursively using a graphing calculator: • On the home screen, key in the value for the initial term \(a_{1}\) and press [ENTER]. • Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes \([2 \mathrm{ND}]\) ANS for the previous term \(a_{n-1}.\) Press \([\mathrm{ENTER}].\) • Continue pressing \([\text { ENTER }]\) to calculate the values for each successive term. Use the steps above to find the indicated term or terms for the sequence. Find the \(15^{\text { th }}\) term of the sequence \(a_{1}=625, a_{n}=0.8 a_{n-1}+18.\)

Use this data for the exercises that follow: In 2013 , there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over). \(^{[34]}\) If you meet five U.S. citizens, what is the percent chance that four are elderly? (Round to the nearest thousandth of a percent.)

A boulder rolled down a mountain, traveling 6 feet in the fi st second. Each successive second, its distance increased by 8 feet. How far did the boulder travel after 10 seconds?

For the following exercises, use the information provided to graph the first 5 terms of the arithmetic sequence. $$ a_{1}=9 ; a_{n}=a_{n-1}-10 $$

Follow these steps to evaluate a sequence defined recursively using a graphing calculator: • On the home screen, key in the value for the initial term \(a_{1}\) and press [ENTER]. • Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes \([2 \mathrm{ND}]\) ANS for the previous term \(a_{n-1}.\) Press \([\mathrm{ENTER}].\) • Continue pressing \([\text { ENTER }]\) to calculate the values for each successive term. Use the steps above to find the indicated term or terms for the sequence. Find the first five terms of the sequence \(a_{1}=2, a_{n}=2^{\left[\left(a_{n-1}\right)-1\right]}+1.\)

Use this data for the exercises that follow: In 2013 , there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over). \(^{[34]}\) It is predicted that by 2030 , one in five U.S. citizens will be elderly. How much greater will the chances of meeting an elderly person be at that time? What policy changes do you foresee if these statistics hold true?

A scientist places 50 cells in a petri dish. Every hour, the population increases by \(1.5 \% .\) What will the cell count be after 1 day?

Is it possible for a sequence to be both arithmetic and geometric? If so, give an example.

For the following exercises, use the information provided to graph the first 5 terms of the arithmetic sequence. $$ a_{n}=-12+5 n $$

Follow these steps to evaluate a sequence defined recursively using a graphing calculator: • On the home screen, key in the value for the initial term \(a_{1}\) and press [ENTER]. • Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes \([2 \mathrm{ND}]\) ANS for the previous term \(a_{n-1}.\) Press \([\mathrm{ENTER}].\) • Continue pressing \([\text { ENTER }]\) to calculate the values for each successive term. Use the steps above to find the indicated term or terms for the sequence. Find the first ten terms of the sequence \(a_{1}=8, a_{n}=\frac{\left(a_{n-1}+1\right) !}{a_{n-1} !}.\)

A pendulum travels a distance of 3 feet on its fi st swing. On each successive swing, it travels \(\frac{3}{4}\) the distance of the previous swing. What is the total distance traveled by the pendulum when it stops swinging?

For the following exercises, follow the steps to work with the arithmetic sequence an = 3n ? 2 using a graphing calculator: Press [MODE] Select [SEQ] in the fourth line Select [DOT] in the fifth line Press [ENTER] Press \([\mathbf{Y}=]\) nMin is the first counting number for the sequence. Set \(n \mathrm{Min}=1\) \(u(n)\) is the pattern for the sequence. Set \(u(n)=3 n-2\) \(u(n \mathrm{Min})\) is the first number in the sequence. Set \(u(n \mathrm{M} \mathrm{in})=1\) Press \([2 \mathrm{ND}]\) then \([\text { WINDOW ] to go to TBLSET }\) Set Tbistart \(=1\) Set \(\Delta \mathrm{Tbl}=1\) Set Indpnt: Auto and Depend: Auto Press \([2 \mathrm{ND}]\) then \([\mathrm{GRAPH}]\) to go to the \([\mathrm{TABLEJ}\) What are the first seven terms shown in the column with the heading \(u(n) ?\)

Follow these steps to evaluate a sequence defined recursively using a graphing calculator: • On the home screen, key in the value for the initial term \(a_{1}\) and press [ENTER]. • Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes \([2 \mathrm{ND}]\) ANS for the previous term \(a_{n-1}.\) Press \([\mathrm{ENTER}].\) • Continue pressing \([\text { ENTER }]\) to calculate the values for each successive term. Use the steps above to find the indicated term or terms for the sequence. Find the tenth term of the sequence \(a_{1}=2, a_{n}=n a_{n-1}\)

Rachael deposits \(\$ 1,500\) into a retirement fund each year. The fund earns \(8.2 \%\) annual interest, compounded monthly. If she opened her account when she was 19 years old, how much will she have by the time she is 55 ? How much of that amount will be interest earned?

Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following. • In the home screen, press [2ND] LIST. •Scroll over to OPS and choose "seq(" from the drop down list. Press [ENTER]. • In the line headed “Expr:” type in the explicit formula, using the \([\mathbf{X}, \mathbf{T}, \boldsymbol{\theta}, \boldsymbol{n}]\) button for n \(\cdot\) In the line headed "Variable:" type in the variable used on the previous step. • In the line headed “start:” key in the value of n that begins the sequence. • In the line headed “end:” key in the value of n that ends the sequence. • Press [ENTER] 3 times to return to the home screen. You will see the sequence syntax on the screen. Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Using a TI-83, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose “seq(” from the drop down list. Press [ENTER]. • Enter the items in the order “Expr”, “Variable”, “start”, “end” separated by commas. See the instructions above for the description of each item. • Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary. List the first five terms of the sequence. \(a_{n}=-\frac{28}{9} n+\frac{5}{3}.\)

For the following exercises, follow the steps to work with the arithmetic sequence \(a_{n}=3 n-2\) using a graphing calculator: • Press [MODE] › Select [SEQ] in the fourth line › Select [DOT] in the fi h line › Press [ENTER] • Press [Y=] › nMin is the fi st counting number for the sequence. Set nMin = 1 › u(n) is the pattern for the sequence. Set u(n) = 3n ? 2 › u(nMin) is the fi st number in the sequence. Set u(nMin) = 1 • Press [2ND] then [WINDOW] to go to TBLSET › Set TblStart = 1 › Set ?Tbl = 1 › Set Indpnt: Auto and Depend: Auto • Press [2ND] then [GRAPH] to go to the [TABLE] Press [WINDOW]. Set \(n \operatorname{Min}=1, n \operatorname{Max}=5\), \(x \operatorname{Min}=0, x \operatorname{Max}=6, y \operatorname{Min}=-1,\) and \(y \operatorname{Max}=14\). Then press [GRAPH]. Graph the sequence as it appears on the graphing calculator.

Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following. • In the home screen, press [2ND] LIST. •Scroll over to OPS and choose "seq(" from the drop down list. Press [ENTER]. • In the line headed “Expr:” type in the explicit formula, using the \([\mathbf{X}, \mathbf{T}, \boldsymbol{\theta}, \boldsymbol{n}]\) button for n \(\cdot\) In the line headed "Variable:" type in the variable used on the previous step. • In the line headed “start:” key in the value of n that begins the sequence. • In the line headed “end:” key in the value of n that ends the sequence. • Press [ENTER] 3 times to return to the home screen. You will see the sequence syntax on the screen. Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Using a TI-83, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose “seq(” from the drop down list. Press [ENTER]. • Enter the items in the order “Expr”, “Variable”, “start”, “end” separated by commas. See the instructions above for the description of each item. • Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary. List the first six terms of the sequence. \(a_{n}=\frac{n^{3}-3.5 n^{2}+4.1 n-1.5}{2.4 n}\)

Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following. • In the home screen, press [2ND] LIST. •Scroll over to OPS and choose "seq(" from the drop down list. Press [ENTER]. • In the line headed “Expr:” type in the explicit formula, using the \([\mathbf{X}, \mathbf{T}, \boldsymbol{\theta}, \boldsymbol{n}]\) button for n \(\cdot\) In the line headed "Variable:" type in the variable used on the previous step. • In the line headed “start:” key in the value of n that begins the sequence. • In the line headed “end:” key in the value of n that ends the sequence. • Press [ENTER] 3 times to return to the home screen. You will see the sequence syntax on the screen. Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Using a TI-83, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose “seq(” from the drop down list. Press [ENTER]. • Enter the items in the order “Expr”, “Variable”, “start”, “end” separated by commas. See the instructions above for the description of each item. • Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary. List the first five terms of the sequence. \(a_{n}=\frac{15 n \cdot(-2)^{n-1}}{47}\)

Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following. • In the home screen, press [2ND] LIST. •Scroll over to OPS and choose "seq(" from the drop down list. Press [ENTER]. • In the line headed “Expr:” type in the explicit formula, using the \([\mathbf{X}, \mathbf{T}, \boldsymbol{\theta}, \boldsymbol{n}]\) button for n \(\cdot\) In the line headed "Variable:" type in the variable used on the previous step. • In the line headed “start:” key in the value of n that begins the sequence. • In the line headed “end:” key in the value of n that ends the sequence. • Press [ENTER] 3 times to return to the home screen. You will see the sequence syntax on the screen. Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Using a TI-83, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose “seq(” from the drop down list. Press [ENTER]. • Enter the items in the order “Expr”, “Variable”, “start”, “end” separated by commas. See the instructions above for the description of each item. • Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary. List the first four terms of the sequence. \(a_{n}=5.7^{n}+0.275(n-1) !\)

Give two examples of arithmetic sequences whose \(4^{\text {th }}\) terms are 9 .

Follow these steps to evaluate a finite sequence defined by an explicit formula. Using a TI-84, do the following. • In the home screen, press [2ND] LIST. •Scroll over to OPS and choose "seq(" from the drop down list. Press [ENTER]. • In the line headed “Expr:” type in the explicit formula, using the \([\mathbf{X}, \mathbf{T}, \boldsymbol{\theta}, \boldsymbol{n}]\) button for n \(\cdot\) In the line headed "Variable:" type in the variable used on the previous step. • In the line headed “start:” key in the value of n that begins the sequence. • In the line headed “end:” key in the value of n that ends the sequence. • Press [ENTER] 3 times to return to the home screen. You will see the sequence syntax on the screen. Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Using a TI-83, do the following. • In the home screen, press [2ND] LIST. • Scroll over to OPS and choose “seq(” from the drop down list. Press [ENTER]. • Enter the items in the order “Expr”, “Variable”, “start”, “end” separated by commas. See the instructions above for the description of each item. • Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. Use the steps above to find the indicated terms for the sequence. Round to the nearest thousandth when necessary. List the first six terms of the sequence \(a_{n}=\frac{n !}{n}\)

Give two examples of arithmetic sequences whose \(10^{\mathrm{th}}\) terms are 206

Consider the sequence defined by \(a_{n}=-6-8 n .\) Is \(a_{n}=-421\) a term in the sequence? Verify the result.

Find the \(5^{\text {th }}\) term of the arithmetic sequence \(\\{9 b, 5 b, b, \ldots\\}\).

What term in the sequence \(a_{n}=\frac{n^{2}+4 n+4}{2(n+2)}\) has the value 41 ? Verify the result.

Find the \(11^{\text {th }}\) term of the arithmetic sequence \(\\{3 a-2 b, a+2 b,-a+6 b, \ldots\\}\)

Calculate the first eight terms of the sequences \(a_{n}=\frac{(n+2) !}{(n-1) !}\) and \(b_{n}=n^{3}+3 n^{2}+2 n,\) and then make a conjecture about the relationship between these two sequences.