Suppose you invest $140 a month for 3 years into an account earning 8% compounded monthly. After 3 years, you leave the money, without making additional deposits, in the account for another 30 years. How much will you have in the end?

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To calculate how much you'll have in the end, we need to break this problem into two phases:

1. **Phase 1**: During the first 3 years, you invest $140 per month into an account earning 8% interest compounded monthly.
2. **Phase 2**: After 3 years, you stop making deposits but leave the money to grow for another 30 years with interest compounded monthly.

### Phase 1: Monthly Investment for 3 Years

We will use the **future value of a series formula** for an account with monthly investments:

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

Where:
- $ P $ is the monthly payment ($140),
- $ r $ is the monthly interest rate (annual interest rate divided by 12),
- $ n $ is the total number of payments (number of months).

Given:
- $ P = 140 $,
- Annual interest rate = 8% → monthly rate $ r = \frac{0.08}{12} = 0.006667 $,
- Time period = 3 years → $ n = 3 \times 12 = 36 $ months.

Now we can calculate the future value after 3 years.

\[
FV = 140 \times \frac{{(1 + 0.006667)^{36} - 1}}{0.006667}
\]

### Phase 2: Compounding for 30 More Years

Now, the money will grow for another 30 years without additional deposits. This is a simple **future value of a lump sum** formula:

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

Where:
- $ PV $ is the future value from Phase 1,
- $ r $ is the monthly interest rate (still 0.006667),
- $ n $ is the number of months for 30 years (30 × 12 = 360 months).

We'll first calculate the future value after the first 3 years and then use that value to find how much the account will grow over the next 30 years.

Let me do these calculations for you.After 3 years of investing $140 per month in an account earning 8% interest compounded monthly, and then leaving the money to grow for another 30 years without additional deposits, the total amount you will have is approximately **$62,060.03**.

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

Here are 5 related questions to expand on this topic:
1. How would the final amount change if the interest rate were 6% instead of 8%?
2. What would happen if you continued making monthly investments for the full 30 years?
3. How does the compounding frequency (e.g., quarterly, annually) affect the final amount?
4. What if you made an initial lump-sum deposit along with the monthly contributions?
5. How would inflation affect the purchasing power of the $62,060 over time?

**Tip:** Compounding interest grows faster when the interest is applied more frequently. Monthly compounding is more beneficial than annual compounding for the same nominal interest rate.

Suppose you invest $140 a month for 3 years into an account earning 8% compounded monthly. After 3 years, you leave the money, without making additional deposits, in the account for another 30 years. How much will you have in the end?

Find out how much you'll have after investing $140 monthly for 3 years at 8% interest, compounded monthly, followed by 30 years of growth without further deposits. This solution involves key financial math concepts like compound interest and future value formulas. Suitable for high school students and beyond.

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