To attend school, Chloe deposits $\$ 400$ at the end of every quarter for two and one-half years. What is the accumulated value of the deposits if interest is $8 \%$ compounded annually? The accumulated value is $\$ \square$. (Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.)

Step 1 ：Calculate the interest rate per quarter by dividing the annual interest rate by 4: \(\frac{0.08}{4} = 0.02\)

Step 2 ：Calculate the number of quarters in two and one-half years: \(2.5 \times 4 = 10\) quarters

Step 3 ：Calculate the accumulated value of each deposit. The first deposit accumulates interest for \(10 - 1 = 9\) quarters, the second deposit accumulates interest for \(10 - 2 = 8\) quarters, and so on. The last deposit does not accumulate any interest

Step 4 ：The accumulated value of each deposit is given by the formula \(\left(1 + \frac{0.08}{4}\right)^n(\$400)\), where n is the number of quarters the deposit accumulates interest

Step 5 ：Add up the accumulated value of all the deposits: \(\left(1 + \frac{0.08}{4}\right)^9(\$400) + \left(1 + \frac{0.08}{4}\right)^8(\$400) + \ldots + \left(1 + \frac{0.08}{4}\right)^1(\$400) + \$400\)

Step 6 ：This is a geometric series with first term \(\left(1 + \frac{0.08}{4}\right)^9(\$400)\), common ratio \(\left(1 + \frac{0.08}{4}\right)\), and 10 terms. The sum of a geometric series is given by the formula \(\frac{a(1 - r^n)}{1 - r}\), where a is the first term, r is the common ratio, and n is the number of terms

Step 7 ：Substitute the values into the formula to get the final answer: \(\frac{\left(1 + \frac{0.08}{4}\right)^9(\$400)(1 - \left(1 + \frac{0.08}{4}\right)^{10})}{1 - \left(1 + \frac{0.08}{4}\right)} \approx \boxed{\$4,579.64}\)