Question

In order to accumulate enough money for a down payment on a house, a couple deposits $300/month into an account paying 6% compounded monthly. If payments are made at the end of each period, how much money will be in the account in 5 years? What formula is required to complete this problem?

Answer 1

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


`\bf \begin{cases} A= \begin{array}{llll} \textit{accumulated amount}\\ \end{array}\\ pymnt=\textit{periodic payments}\to &300\\ r=rate\to 6\%\to (6)/(100)\to &0.06\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\to &12\\ t=years\to &5 \end{cases}`


`\bf A=300\left[ \cfrac{\left( 1+(0.06)/(12) \right)^(12\cdot 5)-1}{(0.06)/(12)} \right]\implies A=300\left[ \cfrac{(1.005)^(60)-1}{0.005} \right] \\\\\\ A=300(\stackrel{\approx}{69.770030509863214)}\implies A\approx 20931.009152958964`