Answer:
Therefore, approximately $1,761.24 should be deposited in the account to have a balance of $5,000 after 31 years and 6 months with a 2.5% interest compounded quarterly.
Explanation:
A is the future value (desired balance) - $5,000 in this case.
P is the principal amount (the amount to be deposited).
r is the annual interest rate (2.5% or 0.025 as a decimal).
n is the number of times the interest is compounded per year (quarterly compounding means n = 4).
t is the number of years (31.5 years in this case).
Substituting the given values into the formula, we can solve for P:
$5,000 = P(1 + 0.025/4)^(4 * 31.5)
To solve this equation, we can divide both sides by (1 + 0.025/4)^(4 * 31.5) to isolate P:
P = $5,000 / (1 + 0.025/4)^(4 * 31.5)
Using a calculator, we can calculate the value of P:
P ≈ $5,000 / (1.00625)^(126)
P ≈ $5,000 / 2.837
P ≈ $1,761.24