Question

Sydney invests $100 every month into an account that pays 0.8% annual interest, compounded monthly. Benny invests $80 every month into an account that pays 2.2% interest, compounded monthly. Determine the amount In Sydney's account after l0 years.

Answer 1

`~~~~~~~~~~~~\stackrel{\textit{\LARGE Sydney}}{\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 & 100\\ r=rate\to 0.8\%\to (0.8)/(100)\dotfill &0.008\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\dotfill &12\\ t=years\dotfill &10 \end{cases}`

`A=100\left[ \cfrac{\left( 1+(0.008)/(12) \right)^(12 \cdot 10)-1}{(0.008)/(12)} \right]\left(1+(0.008)/(12)\right) \\\\\\ A=100\left[ \cfrac{\left( (1501)/(1500) \right)^(120)-1}{(1)/(500)} \right]\left((1501)/(1500)\right) \implies {\Large \begin{array}{llll} A \approx 12497.05 \end{array}}`