Question

Katie deposits $200 every month into a savings account with an APR of 9%. How much money will be in the account after 8 years?

(Please round your answer to the nearest dollar.)

Answer 1

`~~~~~~~~~~~~\stackrel{\textit{payments at the beginning of the period}}{\textit{Future Value of an annuity due}} \\\\ A=pmt\left[ \cfrac{\left( 1+(r)/(n) \right)^(nt)-1}{(r)/(n)} \right]\left(1+(r)/(n)\right)`

`\qquad \begin{cases} A=\textit{accumulated amount} \\ pmt=\textit{periodic payments}\dotfill & 200\\ r=rate\to 9\%\to (9)/(100)\dotfill &0.09\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\dotfill &12\\ t=years\dotfill &8 \end{cases}`

`A=200\left[ \cfrac{\left( 1+(0.09)/(12) \right)^(12 \cdot 8)-1}{(0.09)/(12)} \right]\left(1+(0.09)/(12)\right) \\\\\\ A=200\left[ \cfrac{1.0075^(96)-1}{0.0075} \right](1.0075) \implies \boxed{A \approx 28181}`