Question

At age 30 you deposit $150 at the end of each month into an IRA that pays 4% interest compounded monthly. At age 65, what will the value of the annuity be?

Answer 1

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

`\begin{cases} A= \begin{array}{llll} \textit{accumulated amount}\\ \end{array}\\ pymnt=\textit{periodic payments}\dotfill &150\\ r=rate\to 4\%\to (4)/(100)\dotfill &0.04\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\dotfill &12\\ t=years\dotfill &35 \end{cases}`

`A=150\left[ \cfrac{\left( 1+(0.04)/(12) \right)^(12\cdot 35)-1} {(0.04)/(12)} \right]\implies A= 150\left[ \cfrac{\left( (301)/(300) \right)^(420)~~ - ~~1}{(1)/(300)} \right] \\\\[-0.35em] ~\dotfill\\\\ ~\hfill A\approx 137059.64~\hfill`