To solve these problems, we’ll use both compound interest formulas:

1. For compounding a finite number of times per year:
   $A = P \left(1 + \frac{r}{n}\right)^{nt}$ where:

   • $A$ is the accumulated amount,
   • $P$ is the principal (initial amount),
   • $r$ is the annual interest rate,
   • $n$ is the number of compounding periods per year,
   • $t$ is the time in years.

2. For continuous compounding:
   $A = P e^{rt}$

Given:

• $P = 14490$
• $r = 7.1\% = 0.071$
• $t = 7$ years

Let's calculate each case.

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(a) Compounded Semiannually ( $n = 2$ )

$A = 14490 \left(1 + \frac{0.071}{2}\right)^{2 \times 7}$ $A = 14490 \left(1 + 0.0355\right)^{14}$ $A \approx 14490 \times 1.627$ $A \approx 23558.23$

(b) Compounded Quarterly ( $n = 4$ )

$A = 14490 \left(1 + \frac{0.071}{4}\right)^{4 \times 7}$ $A = 14490 \left(1 + 0.01775\right)^{28}$ $A \approx 14490 \times 1.631$ $A \approx 23617.59$

(c) Compounded Monthly ( $n = 12$ )

$A = 14490 \left(1 + \frac{0.071}{12}\right)^{12 \times 7}$ $A = 14490 \left(1 + 0.0059167\right)^{84}$ $A \approx 14490 \times 1.634$ $A \approx 23657.46$

(d) Compounded Continuously

$A = 14490 \cdot e^{0.071 \cdot 7}$ $A \approx 14490 \cdot e^{0.497}$ $A \approx 14490 \times 1.644$ $A \approx 23800.56$

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Summary of Results

• (a) Compounded Semiannually: $23,558.23
• (b) Compounded Quarterly: $23,617.59
• (c) Compounded Monthly: $23,657.46
• (d) Compounded Continuously: $23,800.56

Would you like further details on any specific part?

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Related Questions

1. How does changing the compounding frequency impact the accumulated amount?
2. What would be the accumulated value if the interest rate were 8% instead of 7.1%?
3. How much more interest is earned with continuous compounding versus semiannual compounding?
4. How would the accumulated amount change for an investment period of 10 years instead of 7?
5. What would be the difference in the accumulated amount if the initial investment was $20,000 instead?

Tip:

The more frequently interest is compounded, the more interest is accumulated, but the differences diminish with very frequent compounding (like monthly to continuous).