You deposit $300 each month into an account earning 8% interest compounded monthly.

a) How much will you have in the account in 25 years?

To solve this problem, we can use the formula for the future value of a series of monthly deposits with compound interest:

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

Where:
- $ A $ is the future value of the account after the deposits and interest,
- $ P $ is the monthly deposit amount (\$300),
- $ r $ is the monthly interest rate (annual rate divided by 12),
- $ n $ is the total number of deposits (number of months),
- $ t $ is the number of years.

### Step 1: Identify the variables
- $ P = 300 $ (monthly deposit),
- Annual interest rate = 8%, so monthly interest rate $ r = \frac{8\%}{12} = \frac{0.08}{12} = 0.00667 $,
- $ n = 25 \times 12 = 300 $ months (for 25 years).

### Step 2: Apply the formula
\[
A = 300 \times \left( \frac{(1 + 0.00667)^{300} - 1}{0.00667} \right)
\]

Now, we can calculate the result:

1. $ (1 + 0.00667)^{300} = (1.00667)^{300} \approx 6.84848 $
2. Subtract 1: $ 6.84848 - 1 = 5.84848 $
3. Divide by the interest rate: $ \frac{5.84848}{0.00667} \approx 876.78 $
4. Multiply by the monthly deposit: $ 300 \times 876.78 \approx 263,034 $

### Final Answer:
You will have approximately **$263,034** in the account after 25 years.

Would you like more details on any step or have further questions?

Here are some related questions:
1. How would the total change if you deposit for 30 years instead?
2. What if the interest rate was 6% instead of 8%?
3. How much of the total amount comes from the interest earned?
4. What would the final amount be with quarterly compounding?
5. How would a one-time lump sum deposit compare to monthly deposits over time?

**Tip:** Increasing the deposit frequency (e.g., biweekly instead of monthly) can slightly increase the final balance due to more frequent compounding.

You deposit $300 each month into an account earning 8% interest compounded monthly. How much will you have in the account in 25 years?

Find out how much you will have after 25 years of depositing $300 monthly at an 8% interest rate, compounded monthly. This solution uses the compound interest formula for series of deposits, demonstrating exponential growth. Suitable for Grades 10-12.

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