Chapter 9, Problem 65RE

$\text{(a)}$

To determine

To prove: The amount that must invest today to cover the $\text{\$}1000$ payment in $n$ years is $1000{(1.08)}^{-n}$ .

Explanation of Solution

Given information:

  $\text{\$}1000$ annually to preferred charity or school. A perpetuity is the name given to this form of gift. A contribution of $a_n$ today will be worth $a_n{(1.08)}^n$ years if you can make $8\text{\%}$ yearly on money.

Calculation:

Let $x$ be the amount invest today. This amount will be worth $x\cdot {1.08}^n$ in $n$ years, but know that it must be $\text{\$}1000$ hence

  $\begin{array}{c}x\cdot {1.08}^n=1000\\ x=\frac{1000}{{1.08}^n}\\ =1000\cdot {1.08}^{-n}\end{array}$

Therefore, the required amount that must invest today to cover the $\text{\$}1000$ payment in $n$ years is $1000{(1.08)}^{-n}$

  $\text{(}$ proved $\text{)}$ .

$\text{(b)}$

To determine

To find: The construct an infinite series.

Answer to Problem 65RE

The answer:

Explanation of Solution

Given information:

  $\text{\$}1000$ annually to preferred charity or school. A perpetuity is the name given to this form of gift. A contribution of $a_n$ today will be worth $a_n{(1.08)}^n$ years if you can make $8\text{\%}$ yearly on money.

Calculation:

To cover the payment of the first year. Must invest,

  $1000\cdot {1.08}^{-1}$

To cover the payment of the second year. must invest,

  $1000\cdot {1.08}^{-2}$

In general, must invest,

  $1000\cdot {1.08}^{-n}$

To cover the payment of the $\text{nth}$ year.

Since the series contains all of the payments in perpetuity as a result of the preceding step, the series is:

  

Therefore, the required construct an infinite series is:

  

$(c)$

To determine

To find: Whether the series converges and the sum and find its represent.

Answer to Problem 65RE

The answer: The series converges and its sum is $\$12500$ . It represents the amount should invest today in order to cover all the payments in the perpetuity.

Explanation of Solution

Given information:

  $\text{\$}1000$ annually to preferred charity or school. A perpetuity is the name given to this form of gift. A contribution of $a_n$ today will be worth $a_n{(1.08)}^n$ years if you can make $8\text{\%}$ yearly on money.

Calculation:

In part $(a)$ is a geometric series with

  $\begin{array}{c}\ddot{a}=1000\\ r={1.08}^{-1}<1\end{array}$

It converges and its sum is:

  

Invest the amount determined in the previous step in order to finance all payments for perpetuity.

Therefore, the required series converges and its sum is $\$12500$ . It represents the amount should invest today in order to cover all the payments in the perpetuity.