Find the amount (future value) of the ordinary annuity. $200 per week for 7.5 years at 2.5% interest per year compounded weekly.


To find the future value of an ordinary annuity with weekly contributions, we need to adjust the formula to account for the weekly compounding of interest. 

### Given Information:
- Weekly contribution ($P$): \$200
- Annual interest rate ($r$): 2.5% or 0.025
- Duration ($t$): 7.5 years
- Compounding frequency: weekly

### Steps to Calculate the Future Value:

1. **Convert the annual interest rate to a weekly rate**:
   - There are 52 weeks in a year.
   - Weekly interest rate ($i$) = $\frac{r}{52} = \frac{0.025}{52}$.

2. **Calculate the total number of contributions**:
   - Total weeks ($n$) = $7.5 \times 52$.

3. **Use the future value formula for an ordinary annuity with weekly compounding**:
   \[
   FV = P \times \frac{(1 + i)^n - 1}{i}
   \]
   where:
   - $P$ is the weekly payment,
   - $i$ is the weekly interest rate,
   - $n$ is the total number of payments.

### Plugging in the Values:

1. **Calculate the weekly interest rate**:
   \[
   i = \frac{0.025}{52} \approx 0.000480769
   \]

2. **Calculate the total number of payments (weeks)**:
   \[
   n = 7.5 \times 52 = 390
   \]

3. **Compute the future value $FV$**:
   \[
   FV = 200 \times \frac{(1 + 0.000480769)^{390} - 1}{0.000480769}
   \]

Let's perform the calculation:

1. Calculate $ (1 + 0.000480769)^{390} $:
   \[
   (1 + 0.000480769)^{390} \approx 1.100893
   \]

2. Compute $ (1 + 0.000480769)^{390} - 1 $:
   \[
   1.100893 - 1 = 0.100893
   \]

3. Divide by the weekly interest rate $i$:
   \[
   \frac{0.100893}{0.000480769} \approx 209.866256
   \]

4. Multiply by the weekly contribution $P$:
   \[
   FV = 200 \times 209.866256 \approx 41973.25
   \]

### Final Answer:

The future value of the ordinary annuity is **$41,973.25**.

### Would you like further details or have any questions?

### Related Questions:
1. What would the future value be if the interest rate were 3% instead of 2.5%?
2. How much total interest is earned over the 7.5 years?
3. If the weekly contribution increased to $250, how would that affect the future value?
4. What is the future value if the contributions were made at the beginning of each week?
5. How does changing the compounding frequency from weekly to monthly affect the future value?

### Tip:
When dealing with compound interest, always ensure you adjust the interest rate and the number of periods to match the compounding frequency.

Learn how to calculate the future value of an ordinary annuity with weekly contributions of $200 over 7.5 years at a 2.5% annual interest rate. Step-by-step solution provided with formulas and calculations.

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