Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742



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


To show:

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


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



Timeline for individual payments.


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.



For 0<n<

Af=n=1R×(1+i)n1 in equation (1),


To determine


To show:

The sum of the infinite series Ap, is given by


To determine


To find:

The money that must be invested now at 10%peryear, to provide an annuity in perpetuity of $5000peryear.

To determine


To find:

The money that must be invested now at 8%peryear, compounded quarterly, to provide an annuity in perpetuity of $3000peryear.

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Chapter 13 Solutions

Show all chapter solutions add
Sect-13.1 P-11ESect-13.1 P-12ESect-13.1 P-13ESect-13.1 P-14ESect-13.1 P-15ESect-13.1 P-16ESect-13.1 P-17ESect-13.1 P-18ESect-13.1 P-19ESect-13.1 P-20ESect-13.1 P-21ESect-13.1 P-22ESect-13.1 P-23ESect-13.1 P-24ESect-13.1 P-25ESect-13.1 P-26ESect-13.1 P-27ESect-13.1 P-28ESect-13.1 P-29ESect-13.1 P-30ESect-13.1 P-31ESect-13.1 P-32ESect-13.1 P-33ESect-13.1 P-34ESect-13.1 P-35ESect-13.1 P-36ESect-13.1 P-37ESect-13.1 P-38ESect-13.1 P-39ESect-13.1 P-40ESect-13.1 P-41ESect-13.1 P-42ESect-13.1 P-43ESect-13.1 P-44ESect-13.1 P-45ESect-13.1 P-46ESect-13.1 P-47ESect-13.1 P-48ESect-13.1 P-49ESect-13.1 P-50ESect-13.1 P-51ESect-13.1 P-52ESect-13.1 P-53ESect-13.1 P-54ESect-13.1 P-55ESect-13.1 P-56ESect-13.1 P-57ESect-13.1 P-58ESect-13.1 P-59ESect-13.1 P-60ESect-13.1 P-61ESect-13.1 P-62ESect-13.1 P-63ESect-13.1 P-64ESect-13.1 P-65ESect-13.1 P-66ESect-13.1 P-67ESect-13.1 P-68ESect-13.1 P-69ESect-13.1 P-70ESect-13.1 P-71ESect-13.1 P-72ESect-13.1 P-73ESect-13.1 P-74ESect-13.1 P-75ESect-13.1 P-76ESect-13.1 P-77ESect-13.1 P-78ESect-13.1 P-79ESect-13.1 P-80ESect-13.1 P-81ESect-13.1 P-82ESect-13.1 P-83ESect-13.1 P-84ESect-13.1 P-85ESect-13.1 P-86ESect-13.1 P-87ESect-13.1 P-88ESect-13.2 P-1ESect-13.2 P-2ESect-13.2 P-3ESect-13.2 P-4ESect-13.2 P-5ESect-13.2 P-6ESect-13.2 P-7ESect-13.2 P-8ESect-13.2 P-9ESect-13.2 P-10ESect-13.2 P-11ESect-13.2 P-12ESect-13.2 P-13ESect-13.2 P-14ESect-13.2 P-15ESect-13.2 P-16ESect-13.2 P-17ESect-13.2 P-18ESect-13.2 P-19ESect-13.2 P-20ESect-13.2 P-21ESect-13.2 P-22ESect-13.2 P-23ESect-13.2 P-24ESect-13.2 P-25ESect-13.2 P-26ESect-13.2 P-27ESect-13.2 P-28ESect-13.2 P-29ESect-13.2 P-30ESect-13.2 P-31ESect-13.2 P-32ESect-13.2 P-33ESect-13.2 P-34ESect-13.2 P-35ESect-13.2 P-36ESect-13.2 P-37ESect-13.2 P-38ESect-13.2 P-39ESect-13.2 P-40ESect-13.2 P-41ESect-13.2 P-42ESect-13.2 P-43ESect-13.2 P-44ESect-13.2 P-45ESect-13.2 P-46ESect-13.2 P-47ESect-13.2 P-48ESect-13.2 P-49ESect-13.2 P-50ESect-13.2 P-51ESect-13.2 P-52ESect-13.2 P-53ESect-13.2 P-54ESect-13.2 P-55ESect-13.2 P-56ESect-13.2 P-57ESect-13.2 P-58ESect-13.2 P-59ESect-13.2 P-60ESect-13.2 P-61ESect-13.2 P-62ESect-13.2 P-63ESect-13.2 P-64ESect-13.2 P-65ESect-13.2 P-66ESect-13.2 P-67ESect-13.2 P-68ESect-13.2 P-69ESect-13.2 P-70ESect-13.2 P-71ESect-13.2 P-72ESect-13.2 P-73ESect-13.2 P-74ESect-13.2 P-75ESect-13.2 P-76ESect-13.2 P-77ESect-13.2 P-78ESect-13.3 P-1ESect-13.3 P-2ESect-13.3 P-3ESect-13.3 P-4ESect-13.3 P-5ESect-13.3 P-6ESect-13.3 P-7ESect-13.3 P-8ESect-13.3 P-9ESect-13.3 P-10ESect-13.3 P-11ESect-13.3 P-12ESect-13.3 P-13ESect-13.3 P-14ESect-13.3 P-15ESect-13.3 P-16ESect-13.3 P-17ESect-13.3 P-18ESect-13.3 P-19ESect-13.3 P-20ESect-13.3 P-21ESect-13.3 P-22ESect-13.3 P-23ESect-13.3 P-24ESect-13.3 P-25ESect-13.3 P-26ESect-13.3 P-27ESect-13.3 P-28ESect-13.3 P-29ESect-13.3 P-30ESect-13.3 P-31ESect-13.3 P-32ESect-13.3 P-33ESect-13.3 P-34ESect-13.3 P-35ESect-13.3 P-36ESect-13.3 P-37ESect-13.3 P-38ESect-13.3 P-39ESect-13.3 P-40ESect-13.3 P-41ESect-13.3 P-42ESect-13.3 P-43ESect-13.3 P-44ESect-13.3 P-45ESect-13.3 P-46ESect-13.3 P-47ESect-13.3 P-48ESect-13.3 P-49ESect-13.3 P-50ESect-13.3 P-51ESect-13.3 P-52ESect-13.3 P-53ESect-13.3 P-54ESect-13.3 P-55ESect-13.3 P-56ESect-13.3 P-57ESect-13.3 P-58ESect-13.3 P-59ESect-13.3 P-60ESect-13.3 P-61ESect-13.3 P-62ESect-13.3 P-63ESect-13.3 P-64ESect-13.3 P-65ESect-13.3 P-66ESect-13.3 P-67ESect-13.3 P-68ESect-13.3 P-69ESect-13.3 P-70ESect-13.3 P-71ESect-13.3 P-72ESect-13.3 P-73ESect-13.3 P-74ESect-13.3 P-75ESect-13.3 P-76ESect-13.3 P-77ESect-13.3 P-78ESect-13.3 P-79ESect-13.3 P-80ESect-13.3 P-81ESect-13.3 P-82ESect-13.3 P-83ESect-13.3 P-84ESect-13.3 P-85ESect-13.3 P-86ESect-13.3 P-87ESect-13.3 P-88ESect-13.3 P-89ESect-13.3 P-90ESect-13.3 P-91ESect-13.3 P-92ESect-13.3 P-93ESect-13.3 P-94ESect-13.3 P-95ESect-13.3 P-96ESect-13.3 P-97ESect-13.3 P-98ESect-13.3 P-99ESect-13.3 P-100ESect-13.3 P-101ESect-13.3 P-102ESect-13.3 P-103ESect-13.3 P-104ESect-13.4 P-1ESect-13.4 P-2ESect-13.4 P-3ESect-13.4 P-4ESect-13.4 P-5ESect-13.4 P-6ESect-13.4 P-7ESect-13.4 P-8ESect-13.4 P-9ESect-13.4 P-10ESect-13.4 P-11ESect-13.4 P-12ESect-13.4 P-13ESect-13.4 P-14ESect-13.4 P-15ESect-13.4 P-16ESect-13.4 P-17ESect-13.4 P-18ESect-13.4 P-19ESect-13.4 P-20ESect-13.4 P-21ESect-13.4 P-22ESect-13.4 P-23ESect-13.4 P-24ESect-13.4 P-25ESect-13.4 P-26ESect-13.4 P-27ESect-13.4 P-28ESect-13.4 P-29ESect-13.4 P-30ESect-13.4 P-31ESect-13.5 P-1ESect-13.5 P-2ESect-13.5 P-3ESect-13.5 P-4ESect-13.5 P-5ESect-13.5 P-6ESect-13.5 P-7ESect-13.5 P-8ESect-13.5 P-9ESect-13.5 P-10ESect-13.5 P-11ESect-13.5 P-12ESect-13.5 P-13ESect-13.5 P-14ESect-13.5 P-15ESect-13.5 P-16ESect-13.5 P-17ESect-13.5 P-18ESect-13.5 P-19ESect-13.5 P-20ESect-13.5 P-21ESect-13.5 P-22ESect-13.5 P-23ESect-13.5 P-24ESect-13.5 P-25ESect-13.5 P-26ESect-13.5 P-27ESect-13.5 P-28ESect-13.5 P-29ESect-13.5 P-30ESect-13.5 P-31ESect-13.5 P-32ESect-13.5 P-33ESect-13.5 P-34ESect-13.5 P-35ESect-13.5 P-36ESect-13.5 P-37ESect-13.5 P-38ESect-13.6 P-1ESect-13.6 P-2ESect-13.6 P-3ESect-13.6 P-4ESect-13.6 P-5ESect-13.6 P-6ESect-13.6 P-7ESect-13.6 P-8ESect-13.6 P-9ESect-13.6 P-10ESect-13.6 P-11ESect-13.6 P-12ESect-13.6 P-13ESect-13.6 P-14ESect-13.6 P-15ESect-13.6 P-16ESect-13.6 P-17ESect-13.6 P-18ESect-13.6 P-19ESect-13.6 P-20ESect-13.6 P-21ESect-13.6 P-22ESect-13.6 P-23ESect-13.6 P-24ESect-13.6 P-25ESect-13.6 P-26ESect-13.6 P-27ESect-13.6 P-28ESect-13.6 P-29ESect-13.6 P-30ESect-13.6 P-31ESect-13.6 P-32ESect-13.6 P-33ESect-13.6 P-34ESect-13.6 P-35ESect-13.6 P-36ESect-13.6 P-37ESect-13.6 P-38ESect-13.6 P-39ESect-13.6 P-40ESect-13.6 P-41ESect-13.6 P-42ESect-13.6 P-43ESect-13.6 P-44ESect-13.6 P-45ESect-13.6 P-46ESect-13.6 P-47ESect-13.6 P-48ESect-13.6 P-49ESect-13.6 P-50ESect-13.6 P-51ESect-13.6 P-52ESect-13.6 P-53ESect-13.6 P-54ESect-13.6 P-55ESect-13.6 P-56ESect-13.6 P-57ESect-13.6 P-58ESect-13.6 P-59ESect-13.CR P-1CCSect-13.CR P-2CCSect-13.CR P-3CCSect-13.CR P-4CCSect-13.CR P-5CCSect-13.CR P-6CCSect-13.CR P-7CCSect-13.CR P-8CCSect-13.CR P-9CCSect-13.CR P-10CCSect-13.CR P-11CCSect-13.CR P-12CCSect-13.CR P-1ESect-13.CR P-2ESect-13.CR P-3ESect-13.CR P-4ESect-13.CR P-5ESect-13.CR P-6ESect-13.CR P-7ESect-13.CR P-8ESect-13.CR P-9ESect-13.CR P-10ESect-13.CR P-11ESect-13.CR P-12ESect-13.CR P-13ESect-13.CR P-14ESect-13.CR P-15ESect-13.CR P-16ESect-13.CR P-17ESect-13.CR P-18ESect-13.CR P-19ESect-13.CR P-20ESect-13.CR P-21ESect-13.CR P-22ESect-13.CR P-23ESect-13.CR P-24ESect-13.CR P-25ESect-13.CR P-26ESect-13.CR P-27ESect-13.CR P-28ESect-13.CR P-29ESect-13.CR P-30ESect-13.CR P-31ESect-13.CR P-32ESect-13.CR P-33ESect-13.CR P-34ESect-13.CR P-35ESect-13.CR P-36ESect-13.CR P-37ESect-13.CR P-38ESect-13.CR P-39ESect-13.CR P-40ESect-13.CR P-41ESect-13.CR P-42ESect-13.CR P-43ESect-13.CR P-44ESect-13.CR P-45ESect-13.CR P-46ESect-13.CR P-47ESect-13.CR P-48ESect-13.CR P-49ESect-13.CR P-50ESect-13.CR P-51ESect-13.CR P-52ESect-13.CR P-53ESect-13.CR P-54ESect-13.CR P-55ESect-13.CR P-56ESect-13.CR P-57ESect-13.CR P-58ESect-13.CR P-59ESect-13.CR P-60ESect-13.CR P-61ESect-13.CR P-62ESect-13.CR P-63ESect-13.CR P-64ESect-13.CR P-65ESect-13.CR P-66ESect-13.CR P-67ESect-13.CR P-68ESect-13.CR P-69ESect-13.CR P-70ESect-13.CR P-71ESect-13.CR P-72ESect-13.CR P-73ESect-13.CR P-74ESect-13.CR P-75ESect-13.CR P-76ESect-13.CR P-77ESect-13.CR P-78ESect-13.CR P-79ESect-13.CR P-80ESect-13.CR P-81ESect-13.CR P-82ESect-13.CR P-83ESect-13.CT P-1CTSect-13.CT P-2CTSect-13.CT P-3CTSect-13.CT P-4CTSect-13.CT P-5CTSect-13.CT P-6CTSect-13.CT P-7CTSect-13.CT P-8CTSect-13.CT P-9CTSect-13.CT P-10CTSect-13.CT P-11CTSect-13.CT P-12CTSect-13.CT P-13CTSect-13.FOM P-1PSect-13.FOM P-2PSect-13.FOM P-3PSect-13.FOM P-4PSect-13.FOM P-5PSect-13.FOM P-6P