To calculate the future value $A$ of an investment with monthly compounding, we can use the compound interest formula:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

where:

• $P = 26,000$ (the principal),
• $r = 0.09$ (annual interest rate as a decimal),
• $n = 12$ (compounded monthly, so 12 times per year),
• $t = 7$ (time in years).

Step-by-Step Calculation:

1. Calculate the monthly interest rate: $\frac{r}{n} = \frac{0.09}{12} = 0.0075$

2. Calculate the total number of compounding periods: $nt = 12 \times 7 = 84$

3. Plug values into the formula: $A = 26,000 \left(1 + 0.0075\right)^{84}$

4. Calculate the expression inside the parentheses: $1 + 0.0075 = 1.0075$

5. Raise 1.0075 to the power of 84: $1.0075^{84} \approx 1.747422051$

6. Multiply by the principal: $A \approx 26,000 \times 1.747422051 \approx 45,432.97$

Final Answer:

$A \approx 45,432.97$

The future value of the investment is approximately $45,432.97.

Would you like more details, or do you have any questions?

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Here are 5 related questions to explore further:

1. What would be the future value if the interest rate increased to 10%?
2. How does changing the compounding frequency to quarterly affect the future value?
3. What is the effective annual rate (EAR) for a 9% nominal rate compounded monthly?
4. How much total interest is earned over the 7 years?
5. How does a change in the investment period, say to 10 years, affect the future value?

Tip: Monthly compounding can yield significantly more than annual compounding due to more frequent application of interest.