Question

How much money (in dollars) should be invested in an account that earns 5% interest, compounded quarterly, to yield $10,000 in 4 years? Round your answer to the nearest cent.

Answer 1

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\dotfill & \$10000\\ 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{quarterly, thus four} \end{array}\dotfill &4\\ t=years\dotfill &4 \end{cases}`

`10000=P\left(1+(0.05)/(4)\right)^(4\cdot 4) \implies 10000=P(1.0125)^(16) \\\\\\ \cfrac{10000}{(1.0125)^(16)}=P\implies 8197.46\approx P`