To solve this problem, let’s go through each part step by step.

Given Information

• Initial population, $P_0 = 70$.
• Recursive rule, $P_n = P_{n-1} + 95$.

This rule means that each step increases the population by 95.

Step 1: Calculate $P_1$ and $P_2$

Using the recursive rule:

1. $P_1 = P_0 + 95$ $P_1 = 70 + 95 = 165$

2. $P_2 = P_1 + 95$ $P_2 = 165 + 95 = 260$

So, $P_1 = 165$ and $P_2 = 260$.

Step 2: Find an Explicit Formula for $P_n$

Since this is an arithmetic sequence (each term increases by 95), we can use the formula for the $n$-th term of an arithmetic sequence: $P_n = P_0 + n \cdot d$ where $d = 95$ (the common difference).

Substituting $P_0 = 70$: $P_n = 70 + 95n$

Step 3: Use the Explicit Formula to Find $P_{100}$

Substitute $n = 100$ into the formula: $P_{100} = 70 + 95 \cdot 100$ $P_{100} = 70 + 9500 = 9570$

Summary of Answers

• $P_1 = 165$
• $P_2 = 260$
• Explicit formula for $P_n$: $P_n = 70 + 95n$
• $P_{100} = 9570$

Would you like further details on any part of this solution?

Related Questions

1. How would the explicit formula change if the initial population were different?
2. What if the population grew by a different rate, such as 120 instead of 95?
3. How does an arithmetic sequence differ from a geometric sequence?
4. Can you find the population at $n = 50$ using the explicit formula?
5. How could this model be modified to include a variable growth rate?

Tip:

For recursive problems, it’s helpful to recognize if they follow a known sequence (like arithmetic or geometric), as it allows for an explicit formula, which is often simpler to work with than the recursive form.