To find the accumulated amount $A$, we can use the formula for compound interest:

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

Where:

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

Given:

• $P = 2400$,
• $r = 2\% = 0.02$,
• $t = 12 \frac{1}{2}$ years = $12.5$ years,
• The interest is compounded semiannually, so $n = 2$ times per year.

Now, let's substitute these values into the formula:

$A = 2400 \left(1 + \frac{0.02}{2}\right)^{2 \times 12.5}$

$A = 2400 \left(1 + 0.01\right)^{25}$

$A = 2400 \left(1.01\right)^{25}$

Now, calculate $(1.01)^{25}$:

$(1.01)^{25} \approx 1.28203723171$

Now, calculate $A$:

$A = 2400 \times 1.28203723171$

$A \approx 3077.69$

Final Answer:

The accumulated amount $A$ is approximately $3077.69.

Would you like more details or have any questions?

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Further Questions:

1. How would the accumulated amount change if the interest were compounded annually instead of semiannually?
2. How does the interest rate affect the accumulated amount over time?
3. What is the difference between simple interest and compound interest?
4. How do you calculate the number of compounding periods in compound interest calculations?
5. What is the significance of compounding frequency in calculating compound interest?

Tip:

When calculating compound interest, the more frequently the interest is compounded, the higher the accumulated amount will be, due to interest being calculated on the previously accumulated interest.