Q62. - 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 $A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt}$

If the principal, interest rate, and number of interest payments are known, what type of function is $A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt}$? Explain your reasoning.

Step-by-Step Solution

Answer

The equation is an exponential growth equation.

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

$A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt}$

If the principal, interest rate, and number of interest payments are known, the function type $A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt}$ is be determined.

Step 2. Explanation .

Rewriting the given equation by replacing the constants:

$\begin{array}{l}A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt}\\ A\left(t\right)=P{\left({\left(1+\frac{r}{n}\right)}^n\right)}^t\\ A\left(t\right)=a{b}^t\end{array}$

The equation is in the general form of an exponential growth.

Step 3. Conclusion .

Hence the equation is an exponential growth equation.