Recently, More Money 4U offered an annuity that pays 5.1 % compounded monthly. If $1,557 is deposited into this annuity every​ month, how much is in the account after 9 ​years? How much of this is​ interest?

To solve this problem, we will use the **future value of an annuity formula** for monthly compounding:

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

Where:
- $ A $ is the future value of the annuity.
- $ P $ is the monthly deposit (in this case, $ P = 1557 $).
- $ r $ is the annual interest rate (5.1%, or $ r = 0.051 $).
- $ n $ is the number of compounding periods per year (since it’s compounded monthly, $ n = 12 $).
- $ t $ is the number of years (9 years, so $ t = 9 $).

### Step 1: Plug the known values into the formula:

\[
A = 1557 \times \frac{(1 + \frac{0.051}{12})^{12 \times 9} - 1}{\frac{0.051}{12}}
\]

### Step 2: Calculate the future value

- $ \frac{0.051}{12} = 0.00425 $ (monthly interest rate)
- $ 12 \times 9 = 108 $ (number of months)
- $ (1 + 0.00425)^{108} = (1.00425)^{108} \approx 1.563473 $
- Subtract 1: $ 1.563473 - 1 = 0.563473 $

Now, calculate the future value:

\[
A = 1557 \times \frac{0.563473}{0.00425} \approx 1557 \times 132.464 \approx 206193.77
\]

So, the future value $ A $ is approximately **$206,193.77**.

### Step 3: Calculate the total amount deposited and interest earned

- Total deposits over 9 years: $ 1557 \times 12 \times 9 = 1557 \times 108 = 168,156 $.
- Interest earned: $ A - \text{Total deposits} = 206,193.77 - 168,156 = 38,037.77 $.

### Final Answer:
- The amount in the account after 9 years is **$206,193.77**.
- The total interest earned is **$38,037.77**.

Do you want further details or have any questions? 

Here are five related questions you might consider:
1. How would the total change if the deposit amount increased to $2,000 per month?
2. What happens to the future value if the interest rate is compounded quarterly instead of monthly?
3. How much would be in the account after 15 years instead of 9 years?
4. How does changing the interest rate to 6% affect the future value and interest earned?
5. What if you wanted to calculate the present value of an annuity for a different scenario?

**Tip**: To maximize annuity growth, consider increasing contributions earlier rather than later since compounding interest accelerates the growth over time.

Discover how to calculate the future value of an annuity with $1,557 monthly deposits at 5.1% interest, compounded monthly, over 9 years. This detailed solution uses the future value of an annuity formula and explores key concepts such as compound interest and total interest earned.

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