https://file.aiquickdraw.com/mathbot/file/1728405131388cx7woek3.jpg

The problem in the image involves calculating the future value of monthly investments using compound interest. Let's break down and solve the questions step by step.

### Given:
- Goal Amount (Future Value, $ FV $) = \$800,000
- Rate of Return ($ r $) = 7.2% annually (or 0.072 as a decimal)
- Compounded monthly: this means we divide the annual rate by 12
- Time period $ t $ is given in years, but since we're dealing with monthly compounding, we multiply the years by 12.

#### Formula to use:
We will use the future value of a series formula:
\[
FV = P \times \frac{(1 + r/n)^{nt} - 1}{r/n}
\]
Where:
- $ FV $ = future value (in this case, \$800,000)
- $ P $ = monthly investment
- $ r $ = annual interest rate (7.2% or 0.072)
- $ n $ = number of times the interest is compounded per year (12 for monthly)
- $ t $ = number of years
- $ nt $ = total number of compounding periods (monthly compounding)

### 1. **How much should you invest each month for 40 years?**
   - $ FV = 800,000 $
   - $ r = 0.072 $
   - $ n = 12 $ (compounded monthly)
   - $ t = 40 $ years

Substitute these values into the formula and solve for $ P $.

### 2. **How much interest will you earn?**
Interest earned is simply:
\[
\text{Interest} = FV - \text{Total contributions}
\]
Where total contributions is $ P \times n \times t $.

### 3. **How much should you invest each month for 20 years?**
   - Use the same formula, but now with $ t = 20 $ years.

### 4. **How much will your savings be worth after 10 years if you deposit the amount calculated for 20 years?**
Here, the future value needs to be calculated for 10 years with the same monthly investment from part 3.

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How much should you invest each month in order to have $800,000 if your rate of return is 7.2% compounded monthly and you want to achieve your goal in 40 years? How much interest will you earn? How much should you invest each month in order to have $800,000 if you want to achieve your goal in 20 years? If you deposit the amount you need to achieve your goal in 20 years, how much will your savings be worth after 10 years?

Calculate how much you need to invest monthly to reach a goal of $800,000 with a 7.2% interest rate compounded monthly over 20 and 40 years. Learn the formula for the future value of a series and how to determine the interest earned. Suitable for college-level finance students or advanced high school learners.

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