Question

If the maximum payment of $5,000 is made into a Individual Retirement Account (IRA) for 25 years at an annual interest rate of 3.2%, determine the amount in the account. Round to the nearest cent.

Answer 1

now, we're assuming the deposits of $5000 are made at the beginning of the year.

`~~~~~~~~~~~~\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)`

`\begin{cases} A=\textit{accumulated amount} \\ pmt=\textit{periodic payments}\dotfill & 5000\\ r=rate\to 3.2\%\to (3.2)/(100)\dotfill &0.032\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=years\dotfill &25 \end{cases}`

`A=5000\left[ \cfrac{\left( 1+(0.032)/(1) \right)^(1 \cdot 25)-1}{(0.032)/(1)} \right]\left(1+(0.032)/(1)\right) \\\\\\ A=5000\left[ \cfrac{\left( 1.032 \right)^( 25)-1}{0.032} \right]\left(1.032\right) \implies A \approx 193148.73`