Question

New grandparents decide to invest $\$200$per month in an annuity for their grandson. The account will pay $5\%$interest per year which is compounded monthly. How much will be in the child’s account at his twenty-first birthday?

Short answer

Amount in child's account will be equal to$\$88868.36$

Step-by-step solution

Step 1. Given information

Amount of $\$200$ is invested per month in an annuity such that their account will pay 5% interest per year which is compounded monthly.

Step 2. Find the amount in child’s account at his twenty-first birthday. 

Let $P,r,n,t$denote amount invested, rate of interest, the number of times the deposit will be made and the interest compounded each year and number of years.

$\begin{array}{rcl}P& =& \$200\\ r& =& 0.05\\ n& =& 12\\ t& =& 21\end{array}$

So,

To find the Annuity formula, use the formula$\begin{array}{rcl}{A}_t& =& \frac{P\left({\left(1+{\displaystyle \frac{r}{n}}\right)}^{nt}-1\right)}{{\displaystyle \frac{r}{n}}}\end{array}$localid="1645377646619" $\begin{array}{rcl}{A}_t& =& \frac{200\left({\left(1+{\displaystyle \frac{0.05}{12}}\right)}^{12\left(21\right)}-1\right)}{{\displaystyle \frac{0.05}{12}}}\\ & =& 48000\left({\left(1+\frac{0.05}{12}\right)}^{252}-1\right)\\ & =& \$88868.36\end{array}$