The problem you've provided represents a recursive sequence defined as:

$P_n = P_{n-1} + 115 \quad \text{with} \quad P_0 = 80$

To find the general term $P_n$, we can observe the pattern. Here's a step-by-step breakdown:

1. $P_0 = 80$
2. $P_1 = P_0 + 115 = 80 + 115 = 195$
3. $P_2 = P_1 + 115 = 195 + 115 = 310$
4. $P_3 = P_2 + 115 = 310 + 115 = 425$
5. And so on.

General Formula:

We can deduce a general form for $P_n$ from the recursive formula. The recurrence relation is arithmetic, with a common difference of 115. For an arithmetic sequence, the general term is given by:

$P_n = P_0 + n \cdot d$

Where:

• $P_0 = 80$ (initial value),
• $d = 115$ (common difference),
• $n$ is the term number.

Thus, the general term for $P_n$ is:

$P_n = 80 + 115n$

Example:

To calculate $P_4$, we use the formula:

$P_4 = 80 + 115 \times 4 = 80 + 460 = 540$

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

Here are 5 related questions to deepen understanding:

1. How would the general formula change if the initial value $P_0$ were different?
2. What if the common difference were negative?
3. How would you solve for $n$ if a specific $P_n$ value is given?
4. Can this sequence be related to any real-world scenarios like savings or investments?
5. How would a geometric progression differ from this arithmetic sequence?

Tip: Recognizing a recursive pattern can often help simplify and find the general term of a sequence.