Example 18) Rhen is about to begin saving for her newborn son’s college education. She
can save $200 per month with her first payment beginning today. She expects to be able
to earn an APR of 7.8% in the stock market. How much will she have saved for college
on her son’s 18th birthday?


To determine how much Rhen will have saved for her son's college education by his 18th birthday, we'll use the formula for the future value of an annuity due. This formula is used because Rhen is making her first payment today, so each payment will have the opportunity to earn interest immediately.

### Formula for Future Value of an Annuity Due:
\[
FV_{\text{due}} = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \times (1 + r)
\]

Where:
- $ P $ is the payment amount per period ($200).
- $ r $ is the periodic interest rate.
- $ n $ is the total number of payments.

### Step 1: Identify the variables
- Monthly payment ($ P $) = $200
- Annual interest rate (APR) = 7.8%
- Monthly interest rate ($ r $) = $\frac{7.8\%}{12} = \frac{0.078}{12} \approx 0.0065 $ (or 0.65%)
- Number of years = 18
- Total number of payments ($ n $) = 18 years $\times$ 12 months/year = 216 months

### Step 2: Plug in the values into the formula
\[
FV_{\text{due}} = 200 \times \left( \frac{(1 + 0.0065)^{216} - 1}{0.0065} \right) \times (1 + 0.0065)
\]

### Step 3: Calculate the future value

I'll calculate it now.Rhen will have saved approximately **$94,548.98** by her son's 18th birthday.

Would you like more details on how this was calculated or have any other questions?

Here are some related questions you might find interesting:
1. What if the APR was 6% instead of 7.8%? How much would she have saved?
2. How does the future value change if Rhen starts saving $300 per month instead?
3. What if Rhen starts saving a year later, but with the same conditions? What would the future value be?
4. How would the savings differ if the payments were made at the end of each month instead of the beginning?
5. How does the total interest earned compare to the total amount contributed over the 18 years?

**Tip:** When saving for long-term goals like college, starting early and contributing consistently can significantly impact the future value of your savings due to compound interest.

Rhen is about to begin saving for her newborn son’s college education. She can save $200 per month with her first payment beginning today. She expects to be able to earn an APR of 7.8% in the stock market. How much will she have saved for college on her son’s 18th birthday?

Discover how much Rhen will save for her son’s college education by contributing $200 per month with an APR of 7.8%. This step-by-step solution utilizes the Future Value of an Annuity Due formula and highlights key financial concepts such as compound interest and annuities. Perfect for Grades 11-12 and early college learners.

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