What lump sum do parents need to deposit in an account earning $5 \%$, compounded monthly, so that it will grow to $\$ 70,000$ for their son's college fund in 17 years? (Round your answer to the nearest cent.)
\$.
Answer
\(\boxed{P = $30,601.47}\). So, the parents need to deposit approximately $30,601.47 in the account.
Step 1 ：Given that the amount of money accumulated after 17 years is $70,000, the annual interest rate is 5% or 0.05 (in decimal), the interest is compounded monthly (12 times a year), we need to find the principal amount (P).
Step 2 ：We use the formula for compound interest, which is \(A = P(1 + \frac{r}{n})^{nt}\).
Step 3 ：We rearrange the formula to solve for P: \(P = \frac{A}{(1 + \frac{r}{n})^{nt}}\).
Step 4 ：Substitute the given values into the formula: \(P = \frac{$70,000}{(1 + \frac{0.05}{12})^{12*17}}\).
Step 5 ：Calculate the value: \(P = \frac{$70,000}{(1 + 0.00416667)^{204}}\).
Step 6 ：Simplify the calculation: \(P = \frac{$70,000}{2.287679247}\).
Step 7 ：\(\boxed{P = $30,601.47}\). So, the parents need to deposit approximately $30,601.47 in the account.
