Question

Your grandparents started putting money away for you when you were born. If they put $100

each month into an account earning 3% compounded monthly, how much will be in the
account when you turn 18?

Answer 1

now, the assumption being that your granpa's are depositing the money at the beginning of the month.

`~~~~~~~~~~~~\stackrel{\textit{deposits 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 deposits}\dotfill & 100\\ r=rate\to 3\%\to (3)/(100)\dotfill &0.03\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\dotfill &12\\ t=years\dotfill &18 \end{cases}`

`A=100\left[ \cfrac{\left( 1+(0.03)/(12) \right)^(12 \cdot 18)-1}{(0.03)/(12)} \right]\left(1+(0.03)/(12)\right) \\\\\\ A=100\left[ \cfrac{(1.0025)^(216)-1}{0.0025} \right](1.0025) \implies \boxed{A \approx 28665.52}`