Calculate the amount of an annuity at the time of the last payment for each section.
A) You deposit $1000 every year for 8 years at 3%/a compounded annually.

B) You pay $200 for 30 months. The interest rate is 6%/a compounded monthly.

To calculate the amount of an annuity at the time of the last payment, we will use the future value of an ordinary annuity formula. This formula is:

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

where:
- $ P $ is the payment amount
- $ r $ is the interest rate per period
- $ n $ is the number of periods

### A) Annual Deposits of $1000 for 8 Years at 3% Interest Compounded Annually

Given:
- $ P = 1000 $
- $ r = \frac{3}{100} = 0.03 $
- $ n = 8 $

Using the formula:
\[ FV = 1000 \times \frac{(1 + 0.03)^8 - 1}{0.03} \]

First, calculate $ (1 + 0.03)^8 $:
\[ (1.03)^8 \approx 1.2668 \]

Now, apply the values:
\[ FV = 1000 \times \frac{1.2668 - 1}{0.03} \]
\[ FV = 1000 \times \frac{0.2668}{0.03} \]
\[ FV = 1000 \times 8.8933 \]
\[ FV \approx 8893.30 \]

So, the future value of the annuity at the time of the last payment is approximately $8893.30.

### B) Monthly Payments of $200 for 30 Months at 6% Interest Compounded Monthly

Given:
- $ P = 200 $
- $ r = \frac{6}{12 \times 100} = \frac{0.06}{12} = 0.005 $
- $ n = 30 $

Using the formula:
\[ FV = 200 \times \frac{(1 + 0.005)^{30} - 1}{0.005} \]

First, calculate $ (1 + 0.005)^{30} $:
\[ (1.005)^{30} \approx 1.1616 \]

Now, apply the values:
\[ FV = 200 \times \frac{1.1616 - 1}{0.005} \]
\[ FV = 200 \times \frac{0.1616}{0.005} \]
\[ FV = 200 \times 32.32 \]
\[ FV \approx 6464.00 \]

So, the future value of the annuity at the time of the last payment is approximately $6464.00.

Would you like further details or have any questions?

Here are some related questions you might want to ask:
1. How do you calculate the present value of an annuity?
2. What is the difference between an ordinary annuity and an annuity due?
3. How does the compounding frequency affect the future value of an annuity?
4. Can you explain the annuity formula in more detail?
5. What are some common applications of annuities in finance?

**Tip:** Always double-check the compounding period and ensure the interest rate and periods align correctly (e.g., annual interest rate with monthly payments should be divided by 12).

Learn how to calculate the future value of annuities based on deposits and payments. Includes detailed calculations for annual deposits and monthly payments with varying interest rates and compounding frequencies.

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