Question

Jenna saves $2,500 per year in an account that earns 10% interest per year, compounded annually. Jenna will have saved in 30 years. Her account balance is a result of Jenna’s annuity payments .

Answer 1

$~~~~~~~~~~~~\textit{Future Value of an ordinary annuity}\\ ~~~~~~~~~~~~(\textit{payments at the end of the period}) \\\\ A=pymnt\left[ \cfrac{\left( 1+\frac{r}{n} \right)^{nt}-1}{\frac{r}{n}} \right]$

$\begin{cases} A= \begin{array}{llll} \textit{accumulated amount}\\ \end{array}\\ pymnt=\textit{periodic payments}\dotfill &2500\\ r=rate\to 10\%\to \frac{10}{100}\dotfill &0.1\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=years\dotfill &30 \end{cases}$

$A=2500\left[ \cfrac{\left( 1+\frac{0.1}{1} \right)^{1\cdot 30}-1}{\frac{0.1}{1}} \right]\implies A=2500\left[ \cfrac{(1.01)^{30}-1}{0.1} \right] \\\\[-0.35em] ~\dotfill\\\\ ~\hfill A\approx 411235.06~\hfill$