Use the ordinary annuity formula
\[
A=\frac{p\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]}{\frac{r}{n}}
\]
After
(Rour
to determine the accumulated amount in the annuity.
\begin{tabular}{|l|l|}
\hline \begin{tabular}{l}
Periodic \\
Deposit
\end{tabular} & $\$ 4000$ at the end of each year \\
\hline Rate & $4 \%$ compounded annually \\
\hline Time & 40 years \\
\hline
\end{tabular}

Step 1 ：Given a periodic deposit of $4000 at the end of each year, an interest rate of 4% compounded annually, and a time period of 40 years, we are asked to find the accumulated amount in an annuity.

Step 2 ：We use the ordinary annuity formula: \[A=\frac{p\left[\left(1+\frac{r}{n}\right)^{n t}-1\right]}{\frac{r}{n}}\]

Step 3 ：Where: \[A\] is the accumulated amount, \[p\] is the periodic deposit, \[r\] is the interest rate, \[n\] is the number of times the interest is compounded per time period, and \[t\] is the time period.

Step 4 ：In this case, \[p = 4000\], \[r = 0.04\] (4% as a decimal), \[n = 1\] (since the interest is compounded annually), and \[t = 40\].

Step 5 ：We substitute these values into the formula to find the accumulated amount: \[A = \frac{4000\left[\left(1+\frac{0.04}{1}\right)^{1 \times 40}-1\right]}{\frac{0.04}{1}}\]

Step 6 ：Solving the above expression, we find that \[A = 380102.0627936657\]

Step 7 ：Rounding to the nearest cent, the accumulated amount in the annuity after 40 years is approximately $380,102.06.

Step 8 ：Final Answer: \(\boxed{380102.06}\)