Find the periodic payment needed to attain the future value of the annuity. Round your answer to the nearest cent.
$\begin{array}{cccc}\text { Future Value } & \text { Rate } & \text { Compounded } & \text { Time } \\ \$ 29,500 & 5.2 \% & \text { Monthly } & 7 \text { years }\end{array}$
The periodic payment needed is $\$ 221.82^{\circledR}$.

Step 1 ：Given the future value (FV) of the annuity is \$29,500, the annual interest rate (r) is 5.2% or 0.052 in decimal, the interest is compounded monthly which means 12 times a year (n = 12), and the time the money is invested for (t) is 7 years.

Step 2 ：We need to find the periodic payment (P). The formula to calculate the future value of an annuity is: \[FV = P \times \frac{(1 + \frac{r}{n})^{nt} - 1}{\frac{r}{n}}\]

Step 3 ：We can rearrange the formula to solve for P: \[P = FV \times \frac{\frac{r}{n}}{(1 + \frac{r}{n})^{nt} - 1}\]

Step 4 ：Substitute the given values into the formula: \[P = 29500 \times \frac{\frac{0.052}{12}}{(1 + \frac{0.052}{12})^{12 \times 7} - 1}\]

Step 5 ：After calculating, we get P = 291.8949268874804

Step 6 ：Rounding to the nearest cent, the periodic payment needed is approximately \$291.89

Step 7 ：\(\boxed{P \approx \$291.89}\) is the final answer.