Q64. - Algebra 2

Question

Suppose you deposit a principal amount of P dollars in a bank account that pays compound interest. If the annual interest rate is r (expressed as a decimal) and the bank makes interest payments n times every year, the amount of money A you would have after t years is given by $A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt}$

Find the account balance after 20 years.

Step-by-Step Solution

Answer

The account balance after 20 years is $2216.72.

Step 1. Given Information.

Given, if a principal amount of P dollars is deposited in a bank account that pays compound interest and the annual interest rate is r (expressed as a decimal) and the bank makes interest payments n times every year, the amount of money A after t years is given by $A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt}$

The equation giving the amount of money after t years if $1000 is deposited into an account paying 4% annual interest compounded quarterly (four times per year) is $A\left(t\right)=1000{\left(1.01\right)}^{4t}$

The account balance after 20 years is to be determined.

Step 2. Explanation .

Since the balance is to be determined after 20 years, $t=20$

Plugging the values in the equation:

$\begin{array}{l}A\left(t\right)=1000{\left(1.01\right)}^{4t}\\ A\left(t\right)=1000{\left(1.01\right)}^{4\left(20\right)}\\ A\left(t\right)=1000{\left(1.01\right)}^{80}\\ A\left(t\right)=1000\left(2.2167\right)\\ A\left(t\right)\approx 2216.72\end{array}$

Step 3. Conclusion .

Hence the account balance after 20 years is $2216.72