The problem provided involves the compound interest formula:

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

Where:

• $A(t)$ is the amount of money accumulated after $t$ years, including interest.
• $P$ is the principal amount (initial deposit).
• $r$ is the annual interest rate (as a decimal).
• $n$ is the number of times the interest is compounded per year.
• $t$ is the time the money is invested for, in years.

Given Information:

• Initial deposit ($P$): $7250
• Interest rate ($r$): 3.7% or 0.037 as a decimal
• Time ($t$): 20 years
• Part (a): Semi-annual compounding, $n = 2$
• Part (b): Weekly compounding, $n = 52$

Let's now solve both parts:

Part (a) Semi-annual compounding:

$A(20) = 7250 \left(1 + \frac{0.037}{2}\right)^{2 \times 20}$ $A(20) = 7250 \left(1 + 0.0185\right)^{40}$ $A(20) = 7250 \left(1.0185\right)^{40}$ Calculating $(1.0185)^{40}$:

$(1.0185)^{40} \approx 2.03914$ $A(20) \approx 7250 \times 2.03914 = 14,793.26$

Rounding to the nearest dollar:

$A(20) \approx 14,793$

Part (b) Weekly compounding:

$A(20) = 7250 \left(1 + \frac{0.037}{52}\right)^{52 \times 20}$ $A(20) = 7250 \left(1 + 0.0007115\right)^{1040}$ $A(20) = 7250 \left(1.0007115\right)^{1040}$ Calculating $(1.0007115)^{1040}$:

$(1.0007115)^{1040} \approx 2.03899$ $A(20) \approx 7250 \times 2.03899 = 14,792.65$

Rounding to the nearest dollar:

$A(20) \approx 14,793$

Final Answers:

• Part (a): The account will be worth $14,793 in 20 years with semi-annual compounding.
• Part (b): With weekly compounding, the account would be worth $14,793 as well (essentially the same due to rounding).

Would you like more details or have any questions?

Here are five related questions to explore:

1. How does the frequency of compounding affect the final amount?
2. What would the account be worth in 30 years with the same conditions?
3. How would the future value change if the interest rate were 5% instead of 3.7%?
4. What is continuous compounding and how does it compare to weekly compounding?
5. How do different interest rates affect the doubling time of an investment?

Tip: The higher the frequency of compounding, the more often interest is added to the principal, which slightly increases the final amount over time.