You deposit $200 each month into an account earning 3% interest
compounded monthly. How much will you have in the account in 30 years? (Round
the answer to two decimal place accuracy.) show steps

To solve this problem, we can use the **future value of a series formula** for compound interest, as regular monthly deposits are made into the account:

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

where:

- $ FV $ is the future value of the account after all the monthly deposits.
- $ P $ is the monthly deposit.
- $ r $ is the monthly interest rate.
- $ n $ is the total number of deposits (number of months).

### Given:

- Monthly deposit $ P = 200 $
- Annual interest rate $ = 3\% = 0.03 $
- Time period $ = 30 \text{ years} $
- Interest is compounded monthly.

### Steps:

1. **Find the monthly interest rate**:
   \[
   r = \frac{3\%}{12} = \frac{0.03}{12} = 0.0025
   \]

2. **Find the total number of months (deposits)**:
   \[
   n = 30 \times 12 = 360 \text{ months}
   \]

3. **Substitute the values into the future value formula**:
   \[
   FV = 200 \times \left( \frac{(1 + 0.0025)^{360} - 1}{0.0025} \right)
   \]

4. **Calculate the future value**.

I'll compute this now.The future value of the account after 30 years, with monthly deposits of $200 and 3% interest compounded monthly, will be approximately **$116,547.38**.

### Summary of steps:
1. **Monthly interest rate**: $0.0025$
2. **Total number of months**: 360 months
3. **Future Value**: $FV = 200 \times \left( \frac{(1 + 0.0025)^{360} - 1}{0.0025} \right)$

Would you like further details or have any questions? 

Here are 5 related questions to explore further:

1. How much interest is earned over the 30 years?
2. What if the interest rate was compounded quarterly instead of monthly?
3. How would the future value change if the monthly deposit was $300 instead of $200?
4. What if the time period was 40 years instead of 30 years?
5. How much would you need to deposit monthly to reach $150,000 in 30 years at the same interest rate?

**Tip:** Small changes in the interest rate or deposit amount can significantly affect the future value in long-term savings plans.

You deposit $200 each month into an account earning 3% interest compounded monthly. How much will you have in the account in 30 years? (Round the answer to two decimal place accuracy.) show steps

Discover how to calculate the future value of $200 monthly deposits in an account earning 3% interest compounded monthly over 30 years. This detailed step-by-step solution explains how to use the compound interest formula and covers essential concepts like monthly interest rates and the total number of deposits.

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