Question

If you need $8000 to travel to Europe after you graduate in 4 year. How much would your monthly deposits need to be if your account earns interest at 5% compounded monthly?

Answer 1

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


`\bf \qquad \begin{cases} A= \begin{array}{llll} \textit{accumulated amount}\\ \end{array}\to & \begin{array}{llll} 8000 \end{array}\\ pymnt=\textit{periodic payments}\\ r=rate\to 5\%\to (5)/(100)\to &0.05\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\to &12\\ t=years\to &4 \end{cases}`


`\bf 8000=pymnt\left[ \cfrac{\left( 1+(0.05)/(12) \right)^(12\cdot 4)-1}{(0.05)/(12)} \right] \\\\\\ \cfrac{8000}{\left[ (\left( 1+(0.05)/(12) \right)^(12\cdot 4)-1)/((0.05)/(12)) \right]}=pymnt\implies \cfrac{8000}{(\left( (241)/(240) \right)^(48)-1)/((1)/(240))}=pymnt \\\\\\ \cfrac{8000}{53.0148852}\approx pymnt\implies 150.9010152318\approx pymnt`