   Chapter 4.6, Problem 35E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Effective Rate The effective rate of interest r e f f is the annual rate that will produce the same interest per year as the nominal rate r.(a) For a rate r (in decimal form) that is compounded n times per year, show that the effective rate r e f f (in decimal form) is r e f f = ( 1 + r n ) n − 1 (b) For a rate r (in decimal form) that is compounded continuously, show that the effective rate r e f f (in decimal form) is r e f f = e r —   1 .

(a)

To determine

To prove: The effective rate of interest reff that will produce the same interest per year as the nominal rate r compounded n times per year is given by the formula:

reff=(1+rn)n1

Explanation

Given Information:

The provided information is that the effective rate of interest is reff and the nominal rate compounded n times per year is r. The relation between reff and r to be verified is

reff=(1+rn)n1

Formula used:

The accumulated amount A, after time t of a principal amount P invested at a rate of interest reff compounded annually is given by the formula:

A=P(1+reff)t

The accumulated amount A, after time t of a principal amount P invested at a rate of interest r compounded n times per year is given by the formula:

A=P(1+rn)nt

Proof:

Consider a principal amount P is invested for time t at effective rate of interest reff and nominal rate of interest r accumulates to the same amount A.

Now, the accumulated amount of investment P invested for time t at effective rate of interest reff is given by the formula,

A=P(1+reff)t(1)

Also, the accumulated amount A, after time t of a principal amount P invested at a rate of interest r compounded n times per year is given by the formula:

A=

(b)

To determine

To prove: The effective rate of interest reff that will produce the same interest per year as the nominal rate r compounded continuously is given by the formula:

reff=er1

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