Chapter 8, Problem 18CR

(a).

To determine

To calculate: the balance in account after 10 years.

Answer to Problem 18CR

After 10 years balance in the account will be $\underset{\_}{\$3,662.38}$ .

Explanation of Solution

Given information:

The sequence representing balance in the account after $n$ quarters when initial deposited amount is $3000 and rate of interest is 2% compounded quarterly is as follows

  $A_n=3000{\left(1+\frac{0.02}{4}\right)}^n$

Calculation:

Expression of account balance is as follows

  $A_n=3000{\left(1+\frac{0.02}{4}\right)}^n$

Need to calculate balance after 10 years

As $n$ represents number of quarters , so for 10 years , number of quarters is $10\times 4=40$

Calculate fortieth term by substituting $n=40$ .

  $A_{40}=3000{\left(1+\frac{0.02}{4}\right)}^{40}$

  $=3000{\left(1.005\right)}^{40}$

  $=3,662.38$

Hence after 40 quarters that is after 10 years balance in the account will be $\underset{\_}{\$3,662.38}$ .

(b).

To determine

To calculate: the balance in account after 20 years and comparison with balance after 10 years.

Answer to Problem 18CR

After 20 years balance in the account will be $\underset{\_}{\$4471.01}$ which is 1.22 times the balance after 10 years.

Explanation of Solution

Given information:

1. The sequence representing balance in the account after $n$ quarters when initial deposited amount is $3000 and rate of interest is 2% compounded quarterly is as follows

  $A_n=3000{\left(1+\frac{0.02}{4}\right)}^n$

2. Balance after 10 years $A_{40}=\$3662.38$

Calculation:

Expression of account balance is as follows

  $A_n=3000{\left(1+\frac{0.02}{4}\right)}^n$

Need to calculate balance after 20 years

As $n$ represents number of quarters, so for 20 years, number of quarters is $20\times 4=80$

Calculate eightieth term by substituting $n=80$ .

  $A_{80}=3000{\left(1+\frac{0.02}{4}\right)}^{80}$

  $=3000{\left(1.005\right)}^{80}$

  $=\$4471.01$

Comparing balance after 20 years and after 10 years.

  $\frac{A_{80}}{A_{40}}=\frac{4471.01}{3662.38}$

  $\frac{A_{80}}{A_{40}}=1.22$

  $A_{80}=1.22A_{40}$

Hence after 80 quarters that is after 20 years balance in the account will be $\underset{\_}{\$4471.01}$ which is 1.22 times the balance after 10 years.