Question

How much money should be invested at 5% interest, compounded quarterly, so that 7 years later the investment will be worth $5000? This amount is called the present value of $5000 at 5% interest.

Answer 1

$\bf \qquad \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\to &\ 5000\\ P=\textit{original amount deposited}\\ r=rate\to 5\%\to \frac{5}{100}\to &0.05\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{quarterly, four times} \end{array}\to &4\\ t=years\to &7 \end{cases} \\\\\\ 5000=P\left(1+\frac{0.05}{4}\right)^{4\cdot 7}\implies 5000=P(1.0125)^{28}\\\\\\ \cfrac{5000}{(1.0125)^{28}}=P$