Chapter 9

The sum of the first \( n \) terms of anarithmetic sequence with first term \( a_1 \) and common difference \( d \) is \( S_n \) Determine the sum if each term is increased by \( 5 \). Explain

In Exercises 113-116, find the indicated partial sum of the series. \( \displaystyle \sum_{i=1}^{\infty} 2 \left(\dfrac{1}{3} \right)^i \) Fifth partial sum

A principal of \(\$ 2500\) is invested at 2\(\%\) interest. Find the amount after 20 years if the interest is compounded (a) annually, (b) semiannually, (c) quarterly, (d) monthly, and (e) daily.

In Exercises 113-116, find the indicated partial sum of the series. \( \displaystyle \sum_{n=1}^{\infty} 4 \left(-\dfrac{1}{2} \right)^n \) Third partial sum

A tool and die company buys a machine for \(\$ 175,000\) and it depreciates at a rate of 30\(\%\) per year. In other words, at the end of each year the depreciated value is 70\(\%\) of what it was at the beginning of the year.) Find the depreciated value of the machine after 5 full years.

In Exercises 113-116, find the indicated partial sum of the series. \( \displaystyle \sum_{n=1}^{\infty} 8 \left(-\dfrac{1}{4} \right)^n \) Fourth partial sum

A deposit of \( \$100 \) is made at the beginning of each month in an account that pays \( 6\% \) interest, compounded monthly. The balance \( A \) in the account at the end of \( 5 \) years is \( A = 100 \left(1 + \dfrac{0.06}{12}\right)^1 + \cdots + 100\left(1 + \dfrac{0.06}{12}\right)^{60} \) Find \( A \).

Exercises 117-120, find the sum of the infinite series. \( \displaystyle \sum_{i=1}^{\infty} 6 \left(\frac{1}{10} \right)^i \)

A deposit of \(50 is made at the beginning of each month in an account that pays 8% interest, compounded monthly. The balance \) A \( in the account at the end of 5 years is \) A = 50 \left(1 + \dfrac{0.08}{12}\right)^1 + \cdots + 50\left(1 + \dfrac{0.06}{12}\right)^{60} \( Find \) A $.

Exercises 117-120, find the sum of the infinite series. \( \displaystyle \sum_{k=1}^{\infty} \left(\frac{1}{10} \right)^k \)

Exercises 117-120, find the sum of the infinite series. \( \displaystyle \sum_{k=1}^{\infty} 7 \left(\frac{1}{10} \right)^k \)

A deposit \( P \) of dollars is made at the beginning of each month in an account with an annual interest rate \( r \) compounded continuously. The balance \( A \) after \( t \) years is \( A = Pe^{r/12} + Pe^{2r/12} + \cdots + Pe^{12tr/12} \). Show that the balance is \( A = \dfrac{Pe^{r/12}\left(e^{rt} - 1\right)}{e^{r/12} - 1} \).

Exercises 117-120, find the sum of the infinite series. \( \displaystyle \sum_{i=1}^{\infty} 2 \left(\frac{1}{10} \right)^i \)

You deposit \( \$25,000 \) in an account that earns \( 7\% \) interest compounded monthly. The balance in the account after \( n \) months is given by \( A_n = 25,000 \left(1 + \dfrac{0.07}{12} \right)^n , n = 1, 2, 3, . . . . \) (a) Write the first six terms of the sequence. (b) Find the balance in the account after 5 years by computing the 60th term of the sequence. (c) Is the balance after 10 years twice the balance after 5 years? Explain.

A deposit of \( \$10,000 \) is made in an account that earns \( 8.5\% \) interest compounded quarterly. The balance in the account after \( n \) quarters is given by \( A_n = 10,000 \left(1 + \dfrac{0.085}{4} \right)^n , n = 1, 2, 3, . . . . \) (a) Write the first eight terms of the sequence. (b) Find the balance in the account after 10 years by computing the 40th term of the sequence. (c) Is the balance after 20 years twice the balance after 10 years? Explain.

The numbers \( a_n \) (in thousands) of AIDS cases reported from 2000 through 2007 can be approximated by the model \( a_n = 0.0768n^3 - 3.150n^2 + 41.56n - 136.4 \) \( n = 10, 11, . . . , 17 \) where \( n \) is the year, with \( n = 10 \) corresponding to 2000. (a) Find the terms of this finite sequence. Use the statistical plotting feature of a graphing utility to construct a bar graph that represents the sequence. (b) What does the graph in part (a) say about reported cases of AIDS?

Consider an initial deposit of \(P\) dollars in an account with an annual interest rate \(r\) , compounded monthly. At the end of each month, a withdrawal of \(W\) dollars will occur and the account will be depleted in \(t\) years. The amount of the initial deposit required is $$\begin{aligned} P=W\left(1+\frac{r}{12}\right)^{-1}+W\left(1+\frac{r}{12}\right)^{-2}+\cdots+& \\\ W\left(1+\frac{r}{12}\right)^{-12 t} & \end{aligned}$$ Show that the initial deposit is $$P=W\left(\frac{12}{r}\right)\left[1-\left(1+\frac{r}{12}\right)^{-12 t}\right]$$

From 1995 to 2007, the federal debt of the United States rose from almost \( \$5 \) trillion to almost \( \$9 \) trillion. The federal debt \( a_n \) (in billions of dollars) from 1995 through 2007 is approximated by the model \( a_n = 1.0904n^3 - 6.348n^2 + 41.76n + 4871.3 \) \( n = 5, 6, . . . , 17 \) where \( n \) is the year, with \( n = 5 \) corresponding to 1995. (a) Find the terms of this finite sequence. Use the statistical plotting feature of a graphing utility to construct a bar graph that represents the sequence. (b) What does the pattern in the bar graph in part (a) say about the future of the federal debt?

In Exercises 127 - 130, use the following information. A tax rebate has been given to property owners by the state government with the anticipation that each property owner will spend approximately \( p\% \) of the rebate, and in turn each recipient of this amount will spend \( p\% \) of what they receive, and so on. Economists refer to this exchange of money and its circulation within the economy as the multiplier effect. The multiplier effect operates on the idea that the expenditures of one individual become the income of another individual. For the given tax rebate, find the total amount put back into the states economy, if this effect continues without end. Tax rebate \( \$400 \) \( p\% \) \( 75\% \)

In Exercises 127 and 128, determine whether the statement is true or false. Justify your answer. \( \displaystyle \sum_{i=1}^{4} (i^2 + 2i) = \sum_{i=1}^{4} i^2 + 2 \sum_{i=1}^{4} i \)

In Exercises 127 - 130, use the following information. A tax rebate has been given to property owners by the state government with the anticipation that each property owner will spend approximately \( p\% \) of the rebate, and in turn each recipient of this amount will spend \( p\% \) of what they receive, and so on. Economists refer to this exchange of money and its circulation within the economy as the multiplier effect. The multiplier effect operates on the idea that the expenditures of one individual become the income of another individual. For the given tax rebate, find the total amount put back into the states economy, if this effect continues without end. Tax rebate \( \$250 \) \( p\% \) \( 80\% \)

In Exercises 127 and 128, determine whether the statement is true or false. Justify your answer. \( \displaystyle \sum_{j=1}^{4} 2^j = \sum_{j=3}^{6} 2^{j - 2} \)

In Exercises 127 - 130, use the following information. A tax rebate has been given to property owners by the state government with the anticipation that each property owner will spend approximately \( p\% \) of the rebate, and in turn each recipient of this amount will spend \( p\% \) of what they receive, and so on. Economists refer to this exchange of money and its circulation within the economy as the multiplier effect. The multiplier effect operates on the idea that the expenditures of one individual become the income of another individual. For the given tax rebate, find the total amount put back into the states economy, if this effect continues without end. Tax rebate \( \$600 \) \( p\% \) \( 72.5\% \)

In Exercises 127 - 130, use the following information. A tax rebate has been given to property owners by the state government with the anticipation that each property owner will spend approximately \( p\% \) of the rebate, and in turn each recipient of this amount will spend \( p\% \) of what they receive, and so on. Economists refer to this exchange of money and its circulation within the economy as the multiplier effect. The multiplier effect operates on the idea that the expenditures of one individual become the income of another individual. For the given tax rebate, find the total amount put back into the states economy, if this effect continues without end. Tax rebate \( \$450 \) \( p\% \) \( 77.5\% \)

In Exercises 131-134, use the following definition of the arithmetic mean \( \bar{x} \) of a set of \( n \) measurements \( x_1, x_2, x_3, \dots , x_n \). \( \displaystyle \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \) Find the arithmetic mean of the six checking account balances \( \$327.15, \$785.69, \$433.04, \$265.38, \$604.12, \) and \( \$590.30 \). Use the statistical capabilities of a graphing utility to verify your result.

In Exercises 131-134, use the following definition of the arithmetic mean \( \bar{x} \) of a set of \( n \) measurements \( x_1, x_2, x_3, \dots , x_n \). \( \displaystyle \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \) Find the arithmetic mean of the following prices per gallon for regular unleaded gasoline at five gasoline stations in a city: \( \$1.899, \$1.959, \$1.919, \$1.939, \) and \( \$1.999 \). Use the statistical capabilities of a graphing utility to verify your result.

An investment firm has a job opening with a salary of \( \$45,000 \) for the first year. Suppose that during the next \( 39 \) years, there is a \( 5\% \) raise each year.Find the total compensation over the \( 40 \)-year period

In Exercises 131-134, use the following definition of the arithmetic mean \( \bar{x} \) of a set of \( n \) measurements \( x_1, x_2, x_3, \dots , x_n \). \( \displaystyle \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \) Prove that \( \displaystyle \sum_{i=1}^{n} (x_i - \bar{x}) = 0 \).

A technology services company has a job opening with a salary of \( \$52,700 \) for the first year.Suppose that during the next \( 24 \) years, there is a \( 3\% \) raise each year. Find the total compensation over the \( 25 \)-year period

In Exercises 131-134, use the following definition of the arithmetic mean \( \bar{x} \) of a set of \( n \) measurements \( x_1, x_2, x_3, \dots , x_n \). \( \displaystyle \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \) Prove that \( \displaystyle \sum_{i=1}^{n} (x_i - \bar{x})^2 = \sum_{i=1}^{n} x_i^2 - \frac{1}{n} \left (\sum_{i=1}^{n} x_i \right)^2 \).

A bungee jumper is jumping off the New River Gorge Bridge in West Virginia, which has a height of \( 876 \) feet. The cord stretches \( 850 \) feet and the jumper rebounds \( 75\% \) of the distance fallen. (a) After jumping and rebounding \( 10 \) times, how far has the jumper traveled downward? How far has the jumper traveled upward? What is the total distance traveled downward and upward? (b) Approximate the total distance, both downward and upward, that the jumper travels before coming to rest.

A ball is dropped from a height of \( 16 \) feet. Each time it drops \( h \) feet, it rebounds \( 0.81 h \) feet. (a) Find the total vertical distance traveled by the ball. (b) The ball takes the following times (in seconds) for each fall. \( s_1 = -16t^2 + 16,\quad \quad \quad \quad \quad s_1 = 0 if t = 1 \) \( s_2 = -16t^2 + 16\left(0.81\right), \quad \quad s_2 = 0 if t = 0.9 \) \( s_3 = -16t^2 + 16\left(0.81\right)^2, \quad \quad s_3 = 0 if t = \left(0.9\right)^2 \) \( s_4 = -16t^2 + 16\left(0.81\right)^3, \quad \quad s_4 = 0 if t = \left(0.9\right)^3 \) \( \vdots \quad \quad \quad \quad \quad \quad \quad\vdots \) \( s_n = -16t^2 + 16\left(0.81\right)^{n - 1}, \quad s_n = 0 if t = \left(0.9\right)^{n-1} \) Beginning with \( s_2 \), the ball takes the same amount of time to bounce up as it does to fall, and so the total time elapsed before it comes to rest is \( t = 1 + 2 \sum_{n=1}^{\infty}\left(0.9\right)^n \) Find this total time.

In Exercises 135-138, find the first five terms of the sequence. \( a_n = \dfrac{(-1)^n x^{2n + 1}}{2n + 1} \)

In Exercises 135-138, find the first five terms of the sequence. \( a_n = \dfrac{(-1)^n x^{2n}}{(2n)!} \)

In Exercises 135-138, find the first five terms of the sequence. \( a_n = \dfrac{(-1)^n x^{2n + 1}}{(2n + 1)!} \)

Consider the graph of \( y = \left(\dfrac{1 - r^x}{1 - r}\right) \). (a) Use a graphing utility to graph \( y \) for \( r = \dfrac{1}{2}, \dfrac{2}{3} \), and \( \dfrac{4}{5} \). What happens as \( x \rightarrow \infty \)? (b) Use a graphing utility to graph \( y \) for \( r = 1.5, 2 \), and \( 3 \). What happens as \( x \rightarrow \infty \) ?

Write out the first five terms of the sequence \(n\)th whose term is \( a_n = \dfrac{(-1)^{n + 1}}{2n + 1} \) Are they the same as the first five terms of the sequence in Example 2? If not, how do they differ?

(a) Write a brief paragraph that describes the similarities and differences between a geometric sequence and a geometric series. Give an example of each. (b) Write a brief paragraph that describes the difference between a finite geometric series and an infinite geometric series. Is it always possible to find the sum of a finite geometric series? Is it always possible to find the sum of an infinite geometric series? Explain.

In your own words, explain the difference between a sequence and a series. Provide examples of each.

Write a brief paragraph explaining why the terms of a geometric sequence decrease in magnitude when \( -1 < r < 1 \).

Find two different geometric series with sums of \( 4 \).