Question: Suppose a friend tells you about an annuity that pays 6% annual interest, compounded semi-annually. You invest in the annuity contributing $10,000 semiannually for 6 years. What is the value of the annuity after your last investment? Enter your answer rounded to the nearest hundred dollars. (For example, $42,570.21 should be entered as 42600).
1. **State the problem:** Calculate the future value of an annuity with semiannual contributions of $10,000, an annual interest rate of 6% compounded semi-annually, over 6 years.
2. **Formula used:** The future value of an ordinary annuity compounded periodically is given by:
$$FV = P \times \frac{(1 + r)^n - 1}{r}$$
where:
- $P$ is the payment per period
- $r$ is the interest rate per period
- $n$ is the total number of payments
3. **Identify values:**
- Annual interest rate = 6% = 0.06
- Compounded semi-annually means 2 compounding periods per year
- Interest rate per period $r = \frac{0.06}{2} = 0.03$
- Number of years = 6
- Number of payments $n = 6 \times 2 = 12$
- Payment per period $P = 10000$
4. **Calculate future value:**
$$FV = 10000 \times \frac{(1 + 0.03)^{12} - 1}{0.03}$$
Calculate $(1 + 0.03)^{12}$:
$$1.03^{12} = 1.42576$$ (approx)
So,
$$FV = 10000 \times \frac{1.42576 - 1}{0.03} = 10000 \times \frac{0.42576}{0.03}$$
5. **Simplify fraction:**
$$\frac{0.42576}{0.03} = 14.192$$ (approx)
6. **Calculate final value:**
$$FV = 10000 \times 14.192 = 141920$$
7. **Round to nearest hundred:**
$$141920 \approx 141900$$
**Final answer:** The value of the annuity after the last investment is approximately $141900$.