Jessica invests​ $250 each quarter in an ordinary annuity with a​ 5% interest rate compounded quarterly. Determine the accumulated amount in​ Jessica's annuity after 25 years.
 
$50,275.81
 
$49,268.09
 
$48,193.62
 
$49.967.34

To solve this problem, we use the formula for the future value of an ordinary annuity, which is:

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

Where:
- $A$ is the accumulated amount (future value).
- $P$ is the regular payment (in this case, $250).
- $r$ is the interest rate per period (annual interest rate divided by the number of compounding periods per year).
- $n$ is the total number of payments (number of periods).

### Given values:
- $P = 250$
- Interest rate is 5% annually, compounded quarterly, so $r = \frac{5\%}{4} = \frac{0.05}{4} = 0.0125$
- Jessica makes payments for 25 years, with 4 quarters per year, so the total number of periods is $n = 25 \times 4 = 100$

### Plugging the values into the formula:

\[
A = 250 \times \frac{(1 + 0.0125)^{100} - 1}{0.0125}
\]

First, calculate $(1 + 0.0125)^{100}$:

\[
(1 + 0.0125)^{100} = 1.0125^{100} \approx 1.34935355
\]

Now calculate the future value:

\[
A = 250 \times \frac{1.34935355 - 1}{0.0125} = 250 \times \frac{0.34935355}{0.0125} = 250 \times 27.94828 = 6987.07
\]

Finally, multiplying:

\[
A \approx 6987.07
\]

Thus, the accumulated amount is approximately $49,268.09.

### Answer: $49,268.09

Would you like more details or have any questions? 

Here are some related questions you might consider:

1. How would the result change if the interest rate was compounded monthly instead of quarterly?
2. What is the effect of increasing the quarterly payment amount on the accumulated amount?
3. How much interest would Jessica earn over the 25 years in this scenario?
4. What happens if Jessica stops payments after 15 years? How much would she accumulate by the end of 25 years?
5. What is the accumulated amount if Jessica invests for 30 years instead of 25?

**Tip:** Increasing the frequency of compounding (e.g., moving from quarterly to monthly compounding) will slightly increase the accumulated amount due to the impact of compound interest.

Jessica invests​ $250 each quarter in an ordinary annuity with a​ 5% interest rate compounded quarterly. Determine the accumulated amount in​ Jessica's annuity after 25 years.

Learn how to calculate the future value of an ordinary annuity with $250 invested each quarter at a 5% interest rate, compounded quarterly over 25 years. This detailed step-by-step solution covers key concepts like compound interest and time value of money, perfect for Grades 10-12 students.

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