Question

Find the amount that should be invested in an account paying 5% compounded monthly so that it will grow to $4000, at the end of 5 years.

a) $5108.43

b) $3141.82

c) $5133.43

d) $3116.82

e) $11,132.78​

Answer 1

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\dotfill & \$ 4000\\ P=\textit{original amount deposited}\\ r=rate\to 5\%\to (5)/(100)\dotfill &0.05\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\dotfill &12\\ t=years\dotfill &5 \end{cases}`

`4000 = P\left(1+(0.05)/(12)\right)^(12\cdot 5) \implies 4000=P\left( (241)/(240) \right)^(60) \\\\\\ \cfrac{4000}{ ~~ \left( (241)/(240) \right)^(60) ~~ }=P\implies 3116.82\approx P`