Find the future value $P$ of the amount $P_{0}=\$ 100,000$ invested for time period $t=7$ years at interest rate $k=5 \%$, compounded continuously. If $\$ 100,000$ is invested, what is the amount accumulated after 7 years? \[ \$ \] (Round to the nearest cent as needed.)

Step 1 ：Given that the principal investment amount \(P_{0} = \$100,000\), the annual interest rate \(k = 5\% = 0.05\), and the time the money is invested for \(t = 7\) years.

Step 2 ：We can use the formula for continuous compounding to find the future value \(P\) of the investment. The formula is given by \[P = P_{0}e^{kt}\]

Step 3 ：Substituting the given values into the formula, we get \[P = 100000 \cdot e^{0.05 \cdot 7}\]

Step 4 ：Calculating the above expression, we find that \(P = 141906.75485932574\)

Step 5 ：Rounding to the nearest cent, we get \(P = \$141,906.75\)

Step 6 ：\(\boxed{The amount accumulated after 7 years is approximately \$141,906.75}\)