Chapter 13.4, Problem 30E

### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

Chapter
Section

### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

# DISCOVER: An Annuity That Lasts Forever An annuity in perpetuity is one that continues forever. Such annuities are useful in setting up scholarship funds to ensure that the award continues.(a) Draw a timeline (as in Example 1) to show that to set up an annuity in perpetuity of amount R per time period, the amount that must be invested now is A p = R 1 + i + R ( 1 + i ) 2 + R ( 1 + i ) 3 + ⋯ + R ( 1 + i ) n + ⋯ where i is the interest rate per time period.(b) Find the sum of the infinite series in part (a) to show that A p = R i (c) How much money must be invested now at 10   %   p e r   y e a r , compounded annually, to provide an annuity in perpetuity of $5000 p e r y e a r ? The first payment is due in 1 y e a r .(d) How much money must be invested now at 8 % p e r y e a r , compounded quarterly, to provide an annuity in perpetuity of$   3000   p e r   y e a r ? The first payment is due in 1   y e a r .

To determine

(a)

To show:

To set up an annuity in perpetuity of amount R per time period, the amount that must be invested now is

Ap=R1+i+R(1+i)2+R(1+i)3++R(1+i)n+

With the help of a timeline, where i is the interest rate per time period.

Explanation

Given:

Timeline for individual payments.

Approach:

The amount invested now to set up an annuity in perpetuity Ap is given by

Ap=Af(1+i)n,0<n< ... (1)

Af is the amount of an annuity,

i is the interest per time period and

n is the total number of installments.

Calculation:

Substitute,

For 0<n<

Af=n=1R×(1+i)n1 in equation (1),

Ap=Af(1+i)n=(n=1R×(1

To determine

(b)

To show:

The sum of the infinite series Ap, is given by

Ap=Ri

To determine

(c)

To find:

The money that must be invested now at 10%peryear, to provide an annuity in perpetuity of $5000peryear. To determine (d) To find: The money that must be invested now at 8%peryear, compounded quarterly, to provide an annuity in perpetuity of$3000peryear.

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