To calculate the accumulated value of an investment for different compounding periods, we use the compound interest formula:

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

Where:

• $A$ is the accumulated value.
• $P$ is the principal amount (initial investment).
• $r$ is the annual interest rate (in decimal form).
• $n$ is the number of times the interest is compounded per year.
• $t$ is the number of years.

Given values:

• $P = 15,000$
• $r = 1.45\% = 0.0145$
• $t = 6$ years

Part a: Compounded Semiannually (n = 2)

For semiannual compounding, $n = 2$.

The formula becomes: $A = 15,000 \left( 1 + \frac{0.0145}{2} \right)^{2 \times 6}$ $A = 15,000 \left( 1 + 0.00725 \right)^{12}$ $A = 15,000 \times (1.00725)^{12}$ $A \approx 15,000 \times 1.093889$ $A \approx 16,358.32$

Part b: Compounded Quarterly (n = 4)

For quarterly compounding, $n = 4$.

The formula becomes: $A = 15,000 \left( 1 + \frac{0.0145}{4} \right)^{4 \times 6}$ $A = 15,000 \left( 1 + 0.003625 \right)^{24}$ $A = 15,000 \times (1.003625)^{24}$ $A \approx 15,000 \times 1.090725$ $A \approx 16,360.88$

Part c: Compounded Monthly (n = 12)

For monthly compounding, $n = 12$.

The formula becomes: $A = 15,000 \left( 1 + \frac{0.0145}{12} \right)^{12 \times 6}$ $A = 15,000 \left( 1 + 0.00120833 \right)^{72}$ $A = 15,000 \times (1.00120833)^{72}$ $A \approx 15,000 \times 1.089625$ $A \approx 16,344.38$

Summary of Results:

• Part a: Accumulated value when compounded semiannually: $16,358.32
• Part b: Accumulated value when compounded quarterly: $16,360.88
• Part c: Accumulated value when compounded monthly: $16,344.38

Would you like further explanation on any of these steps? Here are some related questions to consider:

1. How would the accumulated value change if the interest rate was higher or lower?
2. What would the accumulated value be if the investment period were longer, say 10 years?
3. How does compound interest compare to simple interest in terms of growth?
4. What would happen if the interest were compounded continuously, and how do we calculate that?
5. If the principal amount were doubled, what would be the new accumulated value for monthly compounding?

Tip: The more frequently interest is compounded, the greater the accumulated value, as the interest is applied more often.