Find the future values of the ordinary annuities at the given annual rate r compounded as indicated. The payments are made to coincide with the periods of compounding. (Round your answer to the nearest cent.)
PMT = $100, r = 8.2%, compounded semiannually for 30 years

To find the future value $FV$ of an ordinary annuity with semiannual compounding, we use the formula:

\[
FV = PMT \times \frac{(1 + i)^n - 1}{i}
\]

where:
- $PMT = 100$ (the regular payment),
- $r = 8.2\%$ annual interest rate, which means a semiannual interest rate of $i = \frac{8.2\%}{2} = 4.1\% = 0.041$,
- $n$ is the total number of periods, which is twice the number of years (since it's compounded semiannually). So, $n = 2 \times 30 = 60$.

### Plugging in the values:
\[
FV = 100 \times \frac{(1 + 0.041)^{60} - 1}{0.041}
\]

1. **Calculate $(1 + 0.041)^{60}$:**
   \[
   (1 + 0.041)^{60} \approx 7.20083
   \]

2. **Calculate $(1 + 0.041)^{60} - 1$:**
   \[
   7.20083 - 1 = 6.20083
   \]

3. **Divide by $i$:**
   \[
   \frac{6.20083}{0.041} \approx 151.24
   \]

4. **Multiply by $PMT$:**
   \[
   FV = 100 \times 151.24 = 15124
   \]

So, the future value of the annuity is approximately **$15,124.00**.

Would you like more details on this calculation, or have any questions?

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### Related Questions:
1. How would the future value change if compounding were quarterly instead of semiannually?
2. What would be the future value if the annuity had a 10-year term instead of 30 years?
3. How does changing the payment frequency affect the future value of the annuity?
4. What happens to the future value if the interest rate is increased to 10%?
5. How is the future value affected if payments were made at the beginning of each period (an annuity due)?

#### Tip:
When solving annuity problems, be sure to adjust the interest rate and number of periods to match the compounding frequency.

Find the future values of the ordinary annuities at the given annual rate r compounded as indicated. The payments are made to coincide with the periods of compounding. (Round your answer to the nearest cent.) PMT = $100, r = 8.2%, compounded semiannually for 30 years

Find the future value of an ordinary annuity with semiannual compounding at an 8.2% annual interest rate over 30 years. This problem uses the annuity formula with compound interest calculations, suitable for advanced high school or college-level students.

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