Chapter 7.6, Problem 1MP

(a)

To determine

To find: An expression in terms of r and n for the interest for each compounding period.

Answer to Problem 1MP

  $I=\frac{\mathrm{Pr}}{n}$

Explanation of Solution

The expression for interest is $I=\frac{\mathrm{Pr}}{n}$

Where, P represents the initial deposit

n represents the number of times for which the interest is compounded.

r represents the annual interest rate .

(b)

To determine

To find: An expression in terms of P , r and n for the value of the account after the first compounding period.

Answer to Problem 1MP

  $P_1=P+\frac{\mathrm{Pr}}{n}$

Explanation of Solution

The expression for the value of the account after first compounding period is $P_1=P+\frac{\mathrm{Pr}}{n}$

Where, P represents the initial deposit

n represents the number of times for which the interest is compounded.

r represents the annual interest rate .

(c)

To determine

To explain: The use of distributive property to rewrite the expression in part (b)

Answer to Problem 1MP

  $P_1=P\left(1+\frac{r}{n}\right)$

Explanation of Solution

Given Information: The expression for the value of the account after first compounding period is $P_1=P+\frac{\mathrm{Pr}}{n}$

Formula Used:

Distributive property:

  $a\left(b+c\right)=ab+ac$

Given, $P_1=P+\frac{\mathrm{Pr}}{n}$

Applying Distributive property, i.e., taking the common factor P from the both terms of the Right Hand Side of above equation

  $P_1=P\left(1+\frac{r}{n}\right)$

(d)

To determine

To derive: To derive the formula $A=P{\left(1+\frac{r}{n}\right)}^x$ , which gives the value of the account after x compounding periods.

Explanation of Solution

Given information: The expression for the value of the account after first compounding period is $P_1=P\left(1+\frac{r}{n}\right)$

The expression for the value of the account after first compounding period can be rewrite as

  $P_1=P{\left(1+\frac{r}{n}\right)}^1$.............(1)

Interest for the second compounding period, $I_2=P_1\left(\frac{r}{n}\right)$

Amount after second compounding period,

  $\begin{array}{c}{P}_2=P_1+I_2\\ =P_1+P_1\left(\frac{r}{n}\right)\\ =P_1\left(1+\frac{r}{n}\right)\end{array}$

Substituting (1) to the above equation

  $\begin{array}{c}{P}_2=P\left(1+\frac{r}{n}\right)\left(1+\frac{r}{n}\right)\\ =P{\left(1+\frac{r}{n}\right)}^2\end{array}$

By continuing in this way we get the amount after x compounding period is,

  $A=P{\left(1+\frac{r}{n}\right)}^x$

Verification of the value of CD after one year for Bank West

Given Information:

Amount Emillio wants to invest, $P=15820$

Details of the two banks are shown below:

Bank	CD Length	AnnualInterest Rate	Frequency of compounding
Bank West	6 years	3.8	Quarterly
First Bank	5 years	4.3	Monthly

  $P=\$15820$

  $r=3.8\%$

  $n=4$

  $x=1$

Amount after one year,

  $\begin{array}{c}A=P{\left(1+\frac{r}{n}\right)}^x\\ =15820{\left(1+\frac{3.8}{4\times 100}\right)}^1\\ =\$15970.29\end{array}$

Hence, the proof.

(e)

To determine

To find: To find the value of CD after one year for First Bank

Answer to Problem 1MP

  $A=\$15876.6883$

Explanation of Solution

Given information:

Amount Emillio wants to invest, $P=15820$

Details of the two banks are shown below:

Bank	CD Length	AnnualInterest Rate	Frequency of compounding
Bank West	6 years	3.8	Quarterly
First Bank	5 years	4.3	Monthly

The amount after x compounding period is, $A=P{\left(1+\frac{r}{n}\right)}^x$

Calculation:

  $r=4.3$

  $\begin{array}{l}n=12\\ x=1\end{array}$

  $\begin{array}{c}A=15820{\left(1+\frac{4.3}{12\times 100}\right)}^1\\ =15876.6883\end{array}$

∴The value of CD after one year for First Bank, $A=\$15876.6883$

(f)

To determine

To check whether the formula in (d) is exponential function.

Explanation of Solution

Given information: The amount after x compounding period is, $A=P{\left(1+\frac{r}{n}\right)}^x$

Formula Used:

An exponential function is a function of the form $y=a\cdot b^x$ , where $a\ne 0,b>0,b\ne 1$ and x is a real number.

The given formula for amount $A=P{\left(1+\frac{r}{n}\right)}^x$ is an exponential function since it is expressed in the form $y=a\cdot b^x$ where,

  $a=P\ne 0$

  $b=1+\frac{r}{n}>0$ and $b=1+\frac{r}{n}\ne 1$