John deposits dollar sign, 190$190 every month into an account earning an annual interest rate of 6% compounded monthly. How much would he have in the account after 11 years, *to the nearest dollar*? Use the following formula to determine your answer.

A, equals, d, left parenthesis, start fraction, left parenthesis, 1, plus, i, right parenthesis, to the power n , minus, 1, divided by, i, end fraction, right parenthesis

*A*=*d*(*i*(1+*i*)*n*−1​)

A, equals*A*= the future value of the account after *n* periods
d, equals*d*= the amount invested at the end of each period
i, equals*i*= the interest rate per period
n, equals*n*= the number of periods

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

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

Where:
- $ A $ = the future value of the account after $ n $ periods
- $ d $ = the deposit made at the end of each period
- $ i $ = the interest rate per period
- $ n $ = the total number of periods

We are given:
- $ d = 190 $ (John deposits $190 at the end of each month)
- The annual interest rate is 6%, so the monthly interest rate $ i = \frac{6\%}{12} = 0.06 / 12 = 0.005 $.
- The investment period is 11 years, and since interest is compounded monthly, $ n = 11 \times 12 = 132 $ periods.

Now let's substitute the values into the formula:

\[
A = 190 \left( \frac{{(1 + 0.005)^{132} - 1}}{0.005} \right)
\]

We will now calculate this step by step:

1. Calculate $ 1 + 0.005 = 1.005 $
2. Raise $ 1.005 $ to the power of 132: $ (1.005)^{132} $
3. Subtract 1 from the result.
4. Divide the result by 0.005.
5. Multiply by 190.

After performing these calculations:

\[
A \approx 190 \times 187.1833 \approx 35565.83
\]

So, the future value of the account is approximately **$35,566**.

Would you like a breakdown of each step or any other details?

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Here are 5 questions to expand your understanding:
1. What would happen if John deposited $250 instead of $190 each month?
2. How would the total amount change if the interest rate was 5% instead of 6%?
3. What would be the difference if the compounding period was quarterly instead of monthly?
4. What is the impact of increasing the deposit period to 15 years instead of 11?
5. How would changing the interest rate to be compounded daily affect the future value?

**Tip:** The more frequent the compounding, the greater the future value of the investment due to the power of compounding interest.

John deposits $190 every month into an account earning an annual interest rate of 6% compounded monthly. How much would he have in the account after 11 years, to the nearest dollar? Use the following formula to determine your answer: A = d * [(1 + i)^n - 1] / i.

Learn how to calculate the future value of an annuity with a $190 monthly deposit at an annual interest rate of 6%, compounded monthly, over a period of 11 years. This problem involves the future value of an annuity formula and demonstrates key concepts like compound interest and exponents.

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