A relative has promised to pay you $66.00 today, and he will pay you additional payments every year for the next five years. Each year he will add $67.00 to the previous payment. (So, the payment in year 1 will equal $133.00). You decide to save every dollar you are given and will invest the money in an account paying 6.00% annual interest. How much money will you have accumulated in five years? Keep in mind that you will have six total cash flows to invest.

To solve this problem, we will calculate the future value of each payment you receive, taking into account the 6% annual interest rate. 

### Step-by-Step Solution

#### 1. Determine the Payments:
The payments increase by $67.00 every year, starting from $66.00 today. The payments are as follows:

- **Today (Year 0):** $66.00
- **Year 1:** $133.00
- **Year 2:** $200.00
- **Year 3:** $267.00
- **Year 4:** $334.00
- **Year 5:** $401.00

#### 2. Future Value Formula:
To find out how much each payment will be worth in 5 years, we will use the Future Value (FV) formula for compound interest:

\[
FV = PV \times (1 + r)^n
\]

where:
- $FV$ = Future Value
- $PV$ = Present Value (the amount of each payment)
- $r$ = Interest rate (6% or 0.06)
- $n$ = Number of years the money is invested

#### 3. Calculate the Future Value of Each Payment:

Let's compute the future value of each payment when accumulated over 5 years:

- **Payment Today (Year 0):** 
  \[
  FV_0 = 66 \times (1 + 0.06)^5
  \]

- **Payment in Year 1:** 
  \[
  FV_1 = 133 \times (1 + 0.06)^4
  \]

- **Payment in Year 2:**
  \[
  FV_2 = 200 \times (1 + 0.06)^3
  \]

- **Payment in Year 3:**
  \[
  FV_3 = 267 \times (1 + 0.06)^2
  \]

- **Payment in Year 4:**
  \[
  FV_4 = 334 \times (1 + 0.06)^1
  \]

- **Payment in Year 5:**
  \[
  FV_5 = 401 \times (1 + 0.06)^0
  \]

Now, let's compute each of these future values and sum them up to get the total amount accumulated in five years.The total amount of money accumulated in five years, considering the payments and the 6% annual interest rate, will be approximately **$1,549.48**.

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

### Additional Questions for Exploration:
1. How would the total amount change if the interest rate were 8% instead of 6%?
2. What would be the future value if you received the payments semi-annually instead of annually?
3. How much would you accumulate if you invested the money in an account that compounds interest monthly?
4. What would be the present value of the total amount if discounted back at the same interest rate?
5. How would the future value change if the payments were delayed by one year?

**Tip:** When investing, always consider both the interest rate and the frequency of compounding to maximize your returns.

A relative has promised to pay you $66.00 today, and he will pay you additional payments every year for the next five years. Each year he will add $67.00 to the previous payment. You decide to save every dollar you are given and will invest the money in an account paying 6.00% annual interest. How much money will you have accumulated in five years?

Calculate the future value of increasing payments over five years with a 6% annual interest rate. This problem involves understanding compound interest and future value calculations. Suitable for Grades 10-12, the solution breaks down how to accumulate $1,549.48 in five years through step-by-step analysis.

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