Suppose you invest $160 a month for 4 years into an account earning 6% compounded monthly. After 4 years, you leave the money, without making additional deposits, in the account for another 22 years. How much will you have in the end?


To solve this problem, we can break it into two parts: the accumulation of the investment over the first 4 years, and the growth of that accumulated amount over the next 22 years without additional deposits.

### Part 1: Accumulation During the First 4 Years

You invest $160 per month for 4 years into an account earning 6% interest compounded monthly. 

The future value of a series of regular investments (an annuity) can be calculated using the formula:

\[
FV_1 = P \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- $ P = 160 $ (monthly payment)
- $ r = \frac{0.06}{12} = 0.005 $ (monthly interest rate)
- $ n = 4 \times 12 = 48 $ (total number of payments)

Now, let's calculate $ FV_1 $:

\[
FV_1 = 160 \times \frac{(1 + 0.005)^{48} - 1}{0.005}
\]

### Part 2: Growth Over the Next 22 Years

After 4 years, no additional payments are made, so the amount accumulated will grow for 22 years under the same interest rate.

The future value of a lump sum investment can be calculated using:

\[
FV_2 = FV_1 \times (1 + r)^m
\]

Where:
- $ FV_1 $ is the future value calculated from Part 1
- $ r = 0.005 $ (monthly interest rate)
- $ m = 22 \times 12 = 264 $ (number of months)

Let's compute the final amount $ FV_2 $. I'll calculate this step by step.After 22 years, the final amount in the account will be approximately **$32,295.36**.

Would you like any further details or have any questions?

Here are five related questions you might find interesting:
1. What if the interest rate was compounded quarterly instead of monthly?
2. How would the final amount change if the monthly investment was increased to $200?
3. What if the investment period was extended to 5 years instead of 4 years?
4. How does the future value change if the interest rate was 8% instead of 6%?
5. What would the future value be if you made a one-time investment of $7,680 (equivalent to 4 years of $160 per month) instead of monthly contributions?

**Tip:** Regular contributions to an investment account, even if small, can significantly grow over time due to compound interest.

Suppose you invest $160 a month for 4 years into an account earning 6% compounded monthly. After 4 years, you leave the money, without making additional deposits, in the account for another 22 years. How much will you have in the end?

Discover how to calculate the future value of a $160 monthly investment over 4 years with 6% interest compounded monthly, followed by 22 years of growth without additional deposits. This step-by-step guide covers compound interest and annuity formulas, ideal for advanced high school or college students.

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