To calculate the interest earned on an investment compounded quarterly, we can use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the number of years the money is invested or borrowed.
In this case:
- \( P = 1000 \)
- \( r = 0.08 \) (8% expressed as a decimal)
- \( n = 4 \) (since interest is compounded quarterly)
- \( t = 6 \)
Plugging these values into the formula:
\[ A = 1000 \left(1 + \frac{0.08}{4}\right)^{4 \times 6} \]
Calculating the values inside the brackets:
\[ \frac{0.08}{4} = 0.02 \]
So, we have:
\[ A = 1000 \left(1 + 0.02\right)^{24} \] \[ A = 1000 \left(1.02\right)^{24} \]
Now calculating \( (1.02)^{24} \):
\[ (1.02)^{24} \approx 1.6084 \quad \text{(using a calculator for the exponent)} \]
Now substituting back into the formula for \( A \):
\[ A \approx 1000 \times 1.6084 \approx 1608.40 \]
To find the interest earned, we subtract the principal from the total amount:
\[ \text{Interest} = A - P = 1608.40 - 1000 = 608.40 \]
Thus, the interest earned after 6 years is approximately $608.40.
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