Answer :
To find the maturity value of an investment of [tex]$17,000 growing at an annual interest rate of 6.5%, compounded semi-annually, we can use the compound interest formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the maturity value of the investment.
- \( P \) is the principal amount (initial investment), which is $[/tex]17,000.
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal), which is 0.065.
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year, which is 2 for semi-annual compounding.
- [tex]\( t \)[/tex] is the time the money is invested for, in years, which is 1 year.
Following these steps:
1. Substitute the values into the compound interest formula:
[tex]\[ A = 17000 \left(1 + \frac{0.065}{2}\right)^{2 \times 1} \][/tex]
2. Simplify the expression inside the parentheses:
[tex]\[ A = 17000 \left(1 + 0.0325\right)^2 \][/tex]
[tex]\[ A = 17000 \left(1.0325\right)^2 \][/tex]
3. Calculate the value inside the parentheses raised to the power of 2:
[tex]\[ 1.0325^2 = 1.06655625 \][/tex]
4. Multiply this result by the principal amount:
[tex]\[ A = 17000 \times 1.06655625 \][/tex]
5. Compute the final multiplication to find the maturity value:
[tex]\[ A = 18122.95625 \][/tex]
6. Round the result to the nearest cent:
[tex]\[ A \approx 18122.96 \][/tex]
Therefore, the maturity value of the investment at the end of year 1 is $18,122.96.
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal), which is 0.065.
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year, which is 2 for semi-annual compounding.
- [tex]\( t \)[/tex] is the time the money is invested for, in years, which is 1 year.
Following these steps:
1. Substitute the values into the compound interest formula:
[tex]\[ A = 17000 \left(1 + \frac{0.065}{2}\right)^{2 \times 1} \][/tex]
2. Simplify the expression inside the parentheses:
[tex]\[ A = 17000 \left(1 + 0.0325\right)^2 \][/tex]
[tex]\[ A = 17000 \left(1.0325\right)^2 \][/tex]
3. Calculate the value inside the parentheses raised to the power of 2:
[tex]\[ 1.0325^2 = 1.06655625 \][/tex]
4. Multiply this result by the principal amount:
[tex]\[ A = 17000 \times 1.06655625 \][/tex]
5. Compute the final multiplication to find the maturity value:
[tex]\[ A = 18122.95625 \][/tex]
6. Round the result to the nearest cent:
[tex]\[ A \approx 18122.96 \][/tex]
Therefore, the maturity value of the investment at the end of year 1 is $18,122.96.
