Question: To find the amount of money in Mara's account when she is finished making deposits, use the savings plan formula shown below, where $A$ is the accumulated savings plan balance, $PMT$ is the regular payment amount, $APR$ is the annual percentage rate in decimal form, $n$ is the number of payment periods in a year, and $Y$ is the number of years.
Upper A equals PMT times StartFraction left bracket left parenthesis 1 plus StartFraction APR Over n EndFraction right parenthesis Superscript nY Baseline minus 1 right bracket Over left parenthesis StartFraction APR Over n EndFraction right parenthesis EndFraction
After Mara stops making deposits, the savings plan formula can no longer be used. Instead, use the compound interest formula using the amount of money in her account when she stops making deposits as the principal. The compound interest formula is shown below, where $A$ is the accumulated balance, $P$ is the starting principal, $ARP$ is the annual interest rate in decimal form, and $Y$ is the number of years the interest is compounded.
Upper A equals Upper P times left parenthesis 1 plus APR right parenthesis Superscript Upper Y
1. **State the problem:**
We want to find the amount of money in Mara's account after she finishes making deposits and then after the interest compounds without further deposits.
2. **Savings plan formula:**
The formula to find the accumulated balance $A$ after making regular deposits is:
$$A = PMT \times \frac{\left(1 + \frac{APR}{n}\right)^{nY} - 1}{\frac{APR}{n}}$$
where:
- $PMT$ = regular payment amount
- $APR$ = annual percentage rate (decimal)
- $n$ = number of payment periods per year
- $Y$ = number of years deposits are made
3. **Explanation:**
This formula calculates the future value of an annuity (series of payments) compounded $n$ times per year.
4. **After deposits stop:**
Once Mara stops making deposits, the savings plan formula no longer applies.
Instead, use the compound interest formula:
$$A = P \times (1 + APR)^Y$$
where:
- $P$ = principal amount (the balance when deposits stop)
- $APR$ = annual interest rate (decimal)
- $Y$ = number of years interest is compounded after deposits stop
5. **Summary:**
- Use the savings plan formula to find the balance after deposits.
- Use that balance as $P$ in the compound interest formula to find the final amount after interest compounds without deposits.
This approach correctly models the growth of Mara's savings during and after deposits.