Question: User: Cyrus deposits a fixed quarterly amount into an annuity account for his child's college fund. He wishes to accumulate a future value of $70,000 in 14 years. Assuming an APR of 3.5% compounded quarterly, how much of the $70,000 will Cyrus ultimately deposit in the account, and how much is interest earned? Round your answers to the nearest cent, if necessary.
1. **State the problem:** Cyrus wants to accumulate a future value of $$70000$$ in an annuity account over $$14$$ years with an APR of $$3.5\%$$ compounded quarterly. We need to find the total amount he deposits and the interest earned.
2. **Formula for future value of an ordinary annuity:**
$$
FV = P \times \frac{(1 + r)^n - 1}{r}
$$
where:
- $$FV$$ is the future value ($$70000$$),
- $$P$$ is the quarterly payment (unknown),
- $$r$$ is the interest rate per period,
- $$n$$ is the total number of payments.
3. **Calculate parameters:**
- APR = $$3.5\% = 0.035$$ annually,
- Compounded quarterly means $$4$$ periods per year,
- Interest rate per quarter $$r = \frac{0.035}{4} = 0.00875$$,
- Total number of quarters $$n = 14 \times 4 = 56$$.
4. **Rearrange formula to solve for $$P$$:**
$$
P = \frac{FV \times r}{(1 + r)^n - 1}
$$
5. **Calculate denominator:**
$$
(1 + r)^n - 1 = (1 + 0.00875)^{56} - 1
$$
Calculate $$ (1.00875)^{56} $$:
$$
(1.00875)^{56} \approx 1.601032
$$
So,
$$
1.601032 - 1 = 0.601032
$$
6. **Calculate payment $$P$$:**
$$
P = \frac{70000 \times 0.00875}{0.601032} = \frac{612.5}{0.601032} \approx 1019.23
$$
7. **Calculate total amount deposited:**
$$
\text{Total deposits} = P \times n = 1019.23 \times 56 = 57076.88
$$
8. **Calculate interest earned:**
$$
\text{Interest} = FV - \text{Total deposits} = 70000 - 57076.88 = 12923.12
$$
**Final answers:**
- Cyrus deposits approximately $$57076.88$$ in total.
- Interest earned is approximately $$12923.12$$.