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.