Chapter 14

Find the 15th term of the sequence for which \(a_{n}=-n^{2}-1 . \quad-226\)

For Problems 1–10, find the general term (the nth term) for each sequence. These problems include both arithmetic se- quences and geometric sequences. 3,9,15,21, \ldots, 6 n-3

For Problems 1–10, use mathematical induction to prove each of the sum formulas for the indicated sequences. They are to hold for all positive integers n. S_{n}=\frac{n(n+1)}{2} \quad \text { for } a_{n}=n

Use your knowledge of arithmetic sequences and geometric sequences to help solve Problems 1–28. A man started to work in 1980 at an annual salary of \(\$ 9500\). He received a \(\$ 700\) raise each year. How much was his annual salary in \(2001 ? \quad \$ 24,200\)

For Problems 1–12, find the general term (the nth term) for each geometric sequence. \begin{aligned} &3,6,12,24, \ldots \\ &3(2)^{n-1} \end{aligned}

Write the first five terms of the sequence that has the indicated general term. \(a_{n}=3 n-7\) \(-4,-1,2,5,8\)

Find the fifth term of the sequence for which \(a_{n}=3(2)^{n-1}, \quad 48\)

A woman started to work in 1985 at an annual salary of \(\$ 13,400\). She received a \(\$ 900\) raise each year. How much was her annual salary in 2000? \(\quad \$ 26,900\)

\(2,6,18,54, \ldots\) \(2(3)^{n-1}\)

$$ \begin{aligned} &a_{n}=5 n-2 \\ &8,13,18,23 \end{aligned} $$

Find the general term of the sequence \(-3,-8,-13\), $$ -18, \ldots . $$ $$ -5 n+2 $$

State University had an enrollment of 9600 students in 1992. Each year the enrollment increased by 150 students. What was the enrollment in 2005 ? 11,550

\(3,9,27,81, \ldots \quad 3^{n}\)

3\. Solve \(i=P r t\) for \(t\), given that \(P=\$ 400, r=11 \%\), and \(i=\$ 132\).

Find the general term of the sequence \(5, \frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \ldots .\) \(5(2)^{1-n}\)

Math University had an enrollment of 12,800 students in 1998. Each year the enrollment decreased by 75 students. What was the enrollment in 2005 ? 12,275

$$ \begin{aligned} &a_{n}=-4 n+7 \\ &3,-1,-5,-9,-13 \end{aligned} $$

4\. Solve \(i=P r t\) for \(t\), given that \(P=\$ 250, r=12 \%\), and \(i=\$ 120\).

The enrollment at University \(\mathrm{X}\) is predicted to increase at the rate of \(10 \%\) per year. If the enrollment for 2001 was 5000 students, find the predicted enrollment for 2005 . Express your answer to the nearest whole number.

5\. Solve \(i=P r t\) for \(r\), given that \(P=\$ 600, t=2 \frac{1}{2}\) years, and \(i=\$ 90\). Express \(r\) as a percent.

If you pay \(\$ 12,000\) for a car and it depreciates \(20 \%\) per year, how much will it be worth in 5 years? Express your answer to the nearest dollar. \(\$ 3932\)

\(8,4,2,1, \ldots\) $$ 2^{3}+2^{-n+1}=2^{4-n} $$

$$ \begin{aligned} &a_{n}=2 n^{2}-6 \\ &-4,2,12,26,44 \end{aligned} $$

6\. Solve \(i=P r t\) for \(r\), given that \(P=\$ 700, t=2\) years, and \(i=\$ 126 .\) Express \(r\) as a percent.

Find the 75 th term of the sequence \(1,4,7,10, \ldots\) 223

$$ S_{n}=\frac{n(n+1)(2 n+1)}{6} \text { for } a_{n}=n^{2} $$

A tank contains 16,000 liters of water. Each day one-half of the water in the tank is removed and not replaced. How much water remains in the tank at the end of 7 days? 125 liters

7\. Solve \(i=P r t\) for \(P\), given that \(r=9 \%, t=3\) years, and \(i=\$ 216\).

If the price of a pound of coffee is \(\$ 3.20\) and the projected rate of inflation is \(5 \%\) per year, how much per pound should we expect coffee to cost in 5 years? Express your answer to the nearest cent. $4.08

8\. Solve \(i=P r t\) for \(P\), given that \(r=8 \frac{1}{2} \%, t=2\) years, and \(i=\$ 204\).

A tank contains 5832 gallons of water. Each day onethird of the water in the tank is removed and not replaced. How much water remains in the tank at the end of 6 days? 512 gallons

1,0.3,0.09,0.027, \ldots

9\. Solve \(A=P+P r\) for \(A\), given that \(P=\$ 1000\), \(r=12 \%\), and \(t=5\) years.

A fungus culture growing under controlled conditions doubles in size each day. How many units will the culture contain after 7 days if it originally contains 4 units? 512

10\. Solve \(A=P+P r t\) for \(A\), given that \(P=\$ 850\), \(r=9 \frac{1}{2} \%\), and \(t=10\) years.

Find the sum of the first 45 terms of the sequence for which \(a_{n}=7 n-2 . \quad 7155\)

Find the required term of each of the sequences. The 19 th term of \(1,5,9,13, \ldots\) 73

Use mathematical induction to prove that each statement is true for all positive integers n. $$ 3^{n} \geq 2 n+1 $$

Sue is saving quarters. She saves 1 quarter the first day, 2 quarters the second day, 3 quarters the third day, and so on for 30 days. How much money will she have saved in 30 days? \(\$ 116.25\)

Find the 15 th and 30 th terms of the sequence where $$ a_{n}=-5 n-4 . \quad a_{15}=-79 ; a_{30}=-154 $$

11\. Solve \(A=P+\) Prt for \(r\), given that \(A=\$ 1372\), \(P=\$ 700\), and \(t=12\) years. Express \(r\) as a percent.

Find the sum of the first ten terms of the sequence for which \(a_{n}=3(2)^{n}\).

The 28 th term of \(-2,2,6,10, \ldots\) 106

$$ 4^{n} \geq 4 n $$

Suppose you save a penny the first day of a month, 2 cents the second day, 3 cents the third day, and so on for 31 days. What will be your total savings for the 31 days?

12\. Solve \(A=P+P r t\) for \(r\), given that \(A=\$ 516, P=\$ 300\), and \(t=8\) years. Express \(r\) as a percent.

Find the sum of the first 150 positive even whole numbers. \(\quad 22,650\)

\text { The } 9 \text { th term of } 8,4,2,1, \ldots, \frac{1}{32}

$$ n^{2} \geq n $$

Suppose you save a penny the first day of a month, 2 cents the second day, 4 cents the third day, and continue to double your savings each day. How much will you save on the 15 th day of the month? How much will your total savings be for the 15 days? \(\$ 163.84 ; \$ 327.67\)