Pam and Tim decide to start saving money for their daughter's college education. They open a college savings 529 plan with a $350 initial investment and next month start to make monthly deposits of $110. If the account pays 7% compounded monthly, how much will the account be worth after 180 deposits? Be sure to include the initial investment in the computation. Round the answer to two decimal places.

We are asked to calculate the future value of an investment, which involves both an initial lump sum investment and a series of regular monthly deposits. The account pays interest compounded monthly at a 7% annual rate. 

This is a compound interest problem combined with an annuity calculation, so we can split the problem into two parts:

1. **Initial Investment**: This is a one-time deposit that will grow with compound interest over time.
2. **Monthly Deposits**: These are regular contributions made into the account, which also accrue interest.

### Step 1: Compound Interest Formula for the Initial Investment
The formula for compound interest is:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $ A $ is the future value of the investment.
- $ P $ is the principal amount (the initial investment of $350).
- $ r $ is the annual interest rate (7% or 0.07).
- $ n $ is the number of times the interest is compounded per year (12 times per year for monthly compounding).
- $ t $ is the time in years (180 deposits is 15 years).

We need to calculate the future value of the $350 initial investment after 15 years.

### Step 2: Future Value of Monthly Deposits (Annuity)
The formula for the future value of an annuity (regular deposits) is:

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

Where:
- $ P $ is the monthly deposit ($110).
- $ r $ is the annual interest rate (0.07).
- $ n $ is the number of compounding periods per year (12).
- $ t $ is the number of years (15 years or 180 months total).

Let me now calculate the total future value by combining both the initial investment and the series of monthly deposits.The total value of the account after 180 deposits (15 years), including both the initial investment and the monthly deposits, will be approximately **$35,862.98**.

Would you like any additional details, or do you have any questions?

Here are five related questions you might find interesting:
1. How would the final value change if the interest rate were compounded quarterly instead of monthly?
2. What if Pam and Tim decided to contribute $200 instead of $110 monthly—how would that affect the future value?
3. How much of the final amount comes from interest earned versus total contributions?
4. How would the total amount differ if they saved for 10 years instead of 15?
5. What would be the impact of starting with a larger initial investment?

**Tip:** When planning long-term investments, understanding the impact of compounding frequency and additional contributions can significantly influence your savings growth!

Learn how to calculate the future value of a college savings 529 plan with a $350 initial investment and monthly deposits of $110 at 7% interest, compounded monthly. This detailed step-by-step solution covers compound interest and annuity growth formulas. Suitable for high school students in Grades 11-12.

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