Question

Find the accumulated value of an investment of $25,000 for 7 years at an interest rate of 1.45% if the money is a compounded semiannually; b. compounded quarterly, c.

compounded monthly d. compounded continuously.

Answer 1

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$25000\\ r=rate\to 1.45\%\to (1.45)/(100)\dotfill &0.0145\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{\underline{semi-annually}, thus two} \end{array}\dotfill &2\\ t=years\dotfill &7 \end{cases}`

`A = 25000\left(1+(0.0145)/(2)\right)^(2\cdot 7) \implies A = 25000( 1.00725)^(14)\implies A \approx 27660.62 \\\\[-0.35em] ~\dotfill`

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$25000\\ r=rate\to 1.45\%\to (1.45)/(100)\dotfill &0.0145\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{\underline{quarterly}, thus four} \end{array}\dotfill &4\\ t=years\dotfill &7 \end{cases}`

`A = 25000\left(1+(0.0145)/(4)\right)^(4\cdot 7) \implies A = 25000( 1.003625)^(28)\implies A \approx 27665.67 \\\\[-0.35em] ~\dotfill`

`~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$25000\\ r=rate\to 1.45\%\to (1.45)/(100)\dotfill &0.0145\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{\underline{monthly}, thus twelve} \end{array}\dotfill &12\\ t=years\dotfill &7 \end{cases}`

`A = 25000\left(1+(0.0145)/(12)\right)^(12\cdot 7) \implies A \approx 25000( 1.00121)^(84)\implies A \approx 27669.05 \\\\[-0.35em] ~\dotfill\\\\ ~~~~~~ \textit{\underline{Continuously Compounding} Interest Earned Amount} \\\\ A=Pe^(rt)\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$25000\\ r=rate\to 1.45\%\to (1.45)/(100)\dotfill &0.0145\\ t=years\dotfill &7 \end{cases} \\\\\\ A = 25000e^(0.0145\cdot 7) \implies A \approx 27670.75`