Question

# Use the formula for continuous compounding to compute the balance in the following accounts after $1$, $5$, and $20$ years. Also, find the $\mathrm{APY}$ for each account.A $\ 7000$ deposit in an account with an $\mathrm{APR}$ of $4.5 \%$

Solution

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In the given situation, we need to determine the accumulated values of an investment of $\textcolor{#c34632}{\7,000}$ at an interest rate of $\textcolor{#c34632}{4.5\%}$ compounded continuously after $\textcolor{#c34632}{1}$, $\textcolor{#c34632}{5}$, and $\textcolor{#c34632}{20}$ years.

We will use the compound interest formula for continuous compounding given by:

$A = P \times e^{(APR \times Y)}$

where $A$ is the accumulated balance after $Y$ years, $P$ is the starting principal, $APR$ is the annual percentage rate in decimal form, and $Y$ is the time in years.

Plugging in the known values, we have the following:

After 1 year:

\begin{aligned} A_1 &= \7,000 \times e^{(0.045 \times 1)} \\ &\approx \textcolor{#4257b2}{\7,322.20} \end{aligned}

After 5 years:

\begin{aligned} A_5&= \7,000 \times e^{(0.045 \times 5)} \\ &\approx \textcolor{#4257b2}{\8,766.26} \end{aligned}

After 20 years:

\begin{aligned} A_{20} &= \7,000 \times e^{(0.045 \times 20)} \\ &\approx \textcolor{#4257b2}{\17,217.22} \end{aligned}

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