To find the accumulated amount for an investment using the formula for compound interest, we utilize the formula [tex]\( A = P \left(1 + \frac{r}{n}\right)^{nt} \)[/tex].
Here are the given values:
- Principal amount [tex]\( P \)[/tex] = [tex]$530
- Annual interest rate \( r \) = 8.5% = 0.085 (as a decimal)
- Compounding frequency \( n \) = 2 (since it is compounded semi-annually)
- Time \( t \) = 3 years
We'll apply these values step-by-step into the compound interest formula:
1. Identify the principal amount \( P \):
\( P = 530 \) dollars.
2. Convert the annual interest rate \( r \) from a percentage to a decimal:
\[ r = \frac{8.5}{100} = 0.085 \]
3. Determine the compounding frequency \( n \):
Here, it is compounded semi-annually, so \( n = 2 \).
4. Determine the time in years \( t \):
\( t = 3 \) years.
Now, we substitute these values into the compound interest formula:
\[
A = 530 \left(1 + \frac{0.085}{2}\right)^{2 \times 3}
\]
We can simplify inside the parentheses first:
\[
1 + \frac{0.085}{2} = 1 + 0.0425 = 1.0425
\]
Next, raise this amount to the power of \( 2 \times 3 = 6 \):
\[
A = 530 \left(1.0425\right)^6
\]
Now, calculate \( (1.0425)^6 \):
\[
1.0425^6 \approx 1.28461
\]
Then, multiply this result by the principal amount \( P \):
\[
A = 530 \times 1.28461 \approx 680.343
\]
Finally, we round the result to the nearest cent:
\[
A \approx 680.35
\]
Therefore, the accumulated amount after 3 years, compounded semi-annually at an annual interest rate of 8.5%, is approximately $[/tex]680.35.
