Let's solve each part step-by-step.

Given Information:

• Initial investment: $P = 1000$
• First 4 years: stocks increase at a continuous rate of 5% per year.
• Next 4 years: stocks decrease at a continuous rate of 5% per year.

The formula for continuous compounding is: $A = P e^{rt}$ where:

• $A$ is the amount after time $t$,
• $P$ is the initial principal (investment),
• $r$ is the rate of interest,
• $t$ is the time in years.

Solution:

Part (a): Value of the stocks at the end of the first 4 years

1. For the first 4 years, the rate is 5% (or 0.05 in decimal form).
2. Plugging the values into the formula: $A = 1000 \times e^{0.05 \times 4}$

Calculating $e^{0.05 \times 4}$: $e^{0.05 \times 4} = e^{0.2} \approx 1.2214$ So, $A = 1000 \times 1.2214 \approx 1221.40$

Thus, the value of the stocks at the end of the first 4 years is approximately $1221.40.

Part (b): Value of the stocks at the end of the 8 years

1. Now, after the first 4 years, the stocks start to decrease at a continuous rate of 5% per year for the next 4 years.
2. We take the amount at the end of the first 4 years ($A = 1221.40$) as the new principal and apply the continuous decrease.

So, for the next 4 years: $A = 1221.40 \times e^{-0.05 \times 4}$

Calculating $e^{-0.05 \times 4}$: $e^{-0.05 \times 4} = e^{-0.2} \approx 0.8187$ Thus, $A = 1221.40 \times 0.8187 \approx 1000$

The value of the stocks at the end of 8 years is approximately $1000, which is back to the initial investment amount.

Would you like more details, or do you have any questions?

Here are some related questions for further exploration:

1. What if the interest rate varied annually instead of remaining constant?
2. How would the investment value change if compounding was monthly instead of continuous?
3. What is the effect of a different continuous growth rate, say 3% instead of 5%?
4. How would the outcome differ if the decrease period was longer, say 6 years instead of 4?
5. What is the difference in final value if compounding is discrete rather than continuous?

Tip: When dealing with continuous compounding, remember that the exponential function $e^{rt}$ accurately reflects continuous growth or decay.