Alright! Let's work through this problem step by step:
Initial Investment:
You start with an investment of \[tex]$2,700.
Annual Interest Rate:
The interest rate is 8%, or 0.08 in decimal form.
Number of Years:
The investment period is 3 years.
Let’s address each part of the question sequentially.
### Part a:
What is the value of your investment after one year?
To find the value of the investment after one year, you multiply the initial investment (PV) by the interest rate factor (1 + P).
\[ \text{Value after 1 year} = PV \times (1 + P) \]
\[ \text{Value after 1 year} = 2700 \times 1.08 \]
\[ \text{Value after 1 year} = 2,916.00 \]
So, the value of your investment after one year is \$[/tex]2916.00.
### Part b:
What is the value of your investment after two years?
To find the value after two years, you take the value after one year and multiply it again by the interest rate factor.
[tex]\[ \text{Value after 2 years} = \text{Value after 1 year} \times (1 + P) \][/tex]
[tex]\[ \text{Value after 2 years} = 2916.00 \times 1.08 \][/tex]
[tex]\[ \text{Value after 2 years} = 3,149.28 \][/tex]
Thus, the value of your investment after two years is \[tex]$3,149.28.
### Part c:
What is the value of your investment after three years?
To find the value after three years, take the value after two years and multiply it once more by the interest rate factor.
\[ \text{Value after 3 years} = \text{Value after 2 years} \times (1 + P) \]
\[ \text{Value after 3 years} = 3,149.28 \times 1.08 \]
\[ \text{Value after 3 years} = 3,401.22 \]
So, the value of your investment after three years is \$[/tex]3,401.22.
### Part d:
Combine these steps using the formula [tex]\(FV = PV \times (1 + P)^n\)[/tex] to find the future value of \[tex]$2,700 after 3 years at 8 percent interest.
The future value (FV) can be calculated directly using the future value formula:
\[ FV = PV \times (1 + P)^n \]
Where:
- \(FV\) is the future value of the investment
- \(PV\) is the present value or initial investment (\$[/tex]2,700)
- [tex]\(P\)[/tex] is the annual interest rate (0.08)
- [tex]\(n\)[/tex] is the number of years (3)
Let’s plug in the values:
[tex]\[ FV = 2700 \times (1 + 0.08)^3 \][/tex]
[tex]\[ FV = 2700 \times (1.08)^3 \][/tex]
[tex]\[ FV = 2700 \times 1.259712 \][/tex]
[tex]\[ FV = 3,401.22 \][/tex]
Hence, the future value of your \[tex]$2,700 investment after three years at an 8% interest rate is \$[/tex]3,401.22.
This matches up perfectly with our step-by-step calculations from parts a, b, and c.
