Let's solve this step by step:

Problem (a):

We are given: $p_{t+1} = 0.96p_t \quad \text{and} \quad p_0 = 14$

We need to find $p_{69}$. This is a geometric sequence: $p_t = p_0 \cdot (0.96)^t$

Plugging in $p_0 = 14$ and $t = 69$: $p_{69} = 14 \cdot (0.96)^{69}$

Now, calculate $(0.96)^{69}$ and multiply by 14.

Problem (b):

We are given: $p_{t+1} = p_t + 0.13 \quad \text{and} \quad p_0 = 20$

This is an arithmetic sequence where each term increases by 0.13: $p_t = p_0 + t \cdot 0.13$

We need to find $p_{100}$: $p_{100} = 20 + 100 \cdot 0.13$

Final Calculation:

Let me calculate these values for you. Calculations for Problem (a) and (b)

Problem (a)

p0_a = 14 t_a = 69 r = 0.96 p_69 = p0_a * (r ** t_a)

Problem (b)

p0_b = 20 t_b = 100 increment = 0.13 p_100 = p0_b + t_b * increment

p_69, p_100### Results:

1. Problem (a): $p_{69} = 0.837$ (rounded to three decimal places).
2. Problem (b): $p_{100} = 33.0$.

Would you like me to explain the steps in more detail or have any further questions?

Related Questions:

1. How do geometric sequences differ from arithmetic sequences?
2. How is the formula for the nth term of a geometric sequence derived?
3. What happens to $p_t$ as $t \to \infty$ in part (a)?
4. How can this type of recurrence relation model real-world scenarios?
5. What are the implications of $p_t$ growing linearly in part (b)?

Tip:

When dealing with recurrence relations, always identify if the sequence is arithmetic, geometric, or another form, as this dictates the approach to solving it.