Theodore invests 4720 dollars on a yearly basis and the interest rate is 7.5 percent compounded yearly. He makes the first deposit on June 1, 1973, and the final deposit on June 1, 1995. How much is the investment worth on June 1, 2010 ?


To solve this problem, we need to compute the future value of Theodore's investments using compound interest. Since he makes yearly deposits and the interest is compounded annually, the future value of an annuity formula is appropriate:

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

Where:
- $ P $ is the annual deposit amount.
- $ r $ is the annual interest rate (expressed as a decimal).
- $ n $ is the number of years the investment compounds.

Let's break down the problem step by step:

### Step 1: Time Periods and Interest
- The first deposit is made on June 1, 1973.
- The final deposit is made on June 1, 1995.

Thus, Theodore makes 23 deposits from 1973 to 1995 (inclusive).

- We are calculating the future value as of June 1, 2010. This means the total number of years between June 1, 1995, and June 1, 2010, is $ 2010 - 1995 = 15 $ years.

### Step 2: Calculate the Future Value of Each Deposit
For each deposit, the future value needs to be calculated based on how many years it compounds until June 1, 2010. 

1. The first deposit made on June 1, 1973, compounds for $ 2010 - 1973 = 37 $ years.
2. The second deposit made on June 1, 1974, compounds for $ 2010 - 1974 = 36 $ years.
3. This pattern continues until the final deposit made in 1995, which compounds for $ 2010 - 1995 = 15 $ years.

I will calculate the total future value by summing the future values of all 23 deposits using the formula for compound interest on each deposit. Let me compute that.The total value of Theodore's investment on June 1, 2010, is approximately **$796,445.95**.

Would you like further details on the calculations or have any other questions?

Here are 5 follow-up questions related to this topic:
1. How does changing the interest rate affect the future value of the investment?
2. What would happen if Theodore had deposited monthly instead of yearly?
3. How would the final amount differ if the interest was compounded monthly instead of yearly?
4. What is the effect of delaying the first deposit by one year?
5. How does the future value change if the interest rate fluctuates yearly?

**Tip:** The earlier you start saving or investing, the greater the impact of compound interest over time.

Theodore invests 4720 dollars on a yearly basis and the interest rate is 7.5 percent compounded yearly. He makes the first deposit on June 1, 1973, and the final deposit on June 1, 1995. How much is the investment worth on June 1, 2010?

Explore the future value of Theodore's yearly investments of $4720 from 1973 to 1995 at an interest rate of 7.5% compounded yearly. Detailed step-by-step breakdown using the compound interest formula. Ideal for high school students (Grades 10-12).

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