Find the future value $P$ of the amount $P_{0}$ invested for time period $t$ at interest rate $k$, compounded continuously.
\[
P_{0}=\$ 100,000, t=7 \text { years, } k=3.9 \%
\]
$P=\$ \square$ (Round to the nearest dollar as needed.)
Answer
Calculating the above expression gives the future value of the investment, rounded to the nearest dollar, as \[\boxed{\$128,008}\].
Step 1 ：In the first year, the investment earns \(\frac{0.039}{1}(\$100,000)\) in interest, so the investment is worth \(\$100,000 +\frac{0.039}{1}(\$100,000) = \left(1 + \frac{0.039}{1}\right)(\$100,000)\).
Step 2 ：Similarly, the value of the investment is multiplied by \(1 + \frac{0.039}{1}\) each year, so after 7 years, the investment is worth \[\left(1 + \frac{0.039}{1}\right)^{7}(\$100,000)\].
Step 3 ：Calculating the above expression gives the future value of the investment, rounded to the nearest dollar, as \[\boxed{\$128,008}\].
