You expect to have $292,733  in 11 years. You plan to make regular savings contributions of $X per year for 11 years and a special savings contribution of $24,600 in 3 years. The expected return is 14.80 percent per year and the first regular savings contribution will be made in 1 year. What is X?

This problem involves calculating the regular savings contribution ($ X $) that, along with a special savings contribution and an annual interest rate, will result in a future value of \$292,733 after 11 years. Here's how we can solve it:

We have:
- Future value ($ FV $) = \$292,733
- Number of years ($ N $) = 11 years
- Annual interest rate ($ r $) = 14.80% = 0.148
- Special savings contribution in year 3 = \$24,600

We can break the problem into two parts:
1. The future value of the special contribution of \$24,600 made at the end of year 3.
2. The future value of the annual contributions made at the end of each year for 11 years.

### Step 1: Calculate the future value of the special contribution

The special contribution is made in 3 years and grows for the remaining 8 years (since 11 - 3 = 8 years of growth).

\[
FV_{\text{special}} = 24600 \times (1 + r)^{8}
\]

### Step 2: Calculate the future value of the regular contributions

The regular contributions are an annuity with annual payments $ X $ for 11 years, but with the first payment starting in year 1. The future value of an ordinary annuity is given by:

\[
FV_{\text{annuity}} = X \times \left[\frac{(1 + r)^{N} - 1}{r}\right]
\]

### Step 3: Combine both parts

The total future value is the sum of the future value of the special contribution and the future value of the regular contributions:

\[
FV = FV_{\text{special}} + FV_{\text{annuity}}
\]

Substitute the known values and solve for $ X $.

Let me calculate this now.The regular annual savings contribution ($X$) you need to make is approximately \$9,074.01 per year in order to reach the future value of \$292,733 in 11 years, given the special contribution of \$24,600 in 3 years and an annual return rate of 14.80%.

Would you like more details on the calculation or have any other questions?

Here are 5 related questions for further exploration:
1. How would the regular contribution change if the interest rate were lower?
2. What if the special contribution is made at year 5 instead of year 3?
3. How much interest does the special contribution accrue over 8 years?
4. How would the future value change if contributions were made at the beginning of each year?
5. How would the final amount be affected if you skipped one regular contribution?

**Tip:** The power of compounding works better with more frequent contributions. If possible, consider making smaller, more frequent contributions to maximize growth.

You expect to have $292,733 in 11 years. You plan to make regular savings contributions of $X per year for 11 years and a special savings contribution of $24,600 in 3 years. The expected return is 14.80 percent per year and the first regular savings contribution will be made in 1 year. What is X?

Learn how to calculate the regular annual savings contributions needed to reach a future value of $292,733 in 11 years with a special contribution of $24,600 in 3 years and an annual return rate of 14.80%. This problem involves key financial concepts like the future value of annuities and compound interest. Suitable for advanced high school or undergraduate students.

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