Sure, let's break this down step-by-step using the given initial conditions and recursive rule.
### Step 1: Calculate [tex]\( P_1 \)[/tex] and [tex]\( P_2 \)[/tex]
The initial population is given as:
[tex]\[ P_0 = 70 \][/tex]
The recursive rule for the population is:
[tex]\[ P_n = P_{n-1} + 30 \][/tex]
Using this rule:
1. For [tex]\( P_1 \)[/tex]:
[tex]\[ P_1 = P_0 + 30 \][/tex]
Since [tex]\( P_0 = 70 \)[/tex]:
[tex]\[ P_1 = 70 + 30 = 100 \][/tex]
2. For [tex]\( P_2 \)[/tex]:
[tex]\[ P_2 = P_1 + 30 \][/tex]
Since [tex]\( P_1 = 100 \)[/tex]:
[tex]\[ P_2 = 100 + 30 = 130 \][/tex]
### Step 2: Find an explicit formula for the population
To find the explicit formula, let's observe the pattern in the population growth:
- [tex]\( P_1 = 100 \)[/tex]
- [tex]\( P_2 = 130 \)[/tex]
- Generally, each step increases the population by 30.
This can be generalized as follows:
[tex]\[ P_n = P_0 + n \cdot 30 \][/tex]
Thus, the explicit formula for the population is:
[tex]\[ P_n = 70 + 30n \][/tex]
### Step 3: Use the explicit formula to find [tex]\( P_{100} \)[/tex]
Now we will use the explicit formula to find [tex]\( P_{100} \)[/tex]:
[tex]\[ P_{100} = 70 + 30 \cdot 100 \][/tex]
[tex]\[ P_{100} = 70 + 3000 \][/tex]
[tex]\[ P_{100} = 3070 \][/tex]
To summarize:
[tex]\[
\begin{array}{l}
P_1 = 100 \\
P_2 = 130 \\
\end{array}
\][/tex]
The explicit formula for the population is:
[tex]\[ P_n = 70 + 30n \][/tex]
Finally, the population at [tex]\( n = 100 \)[/tex] is:
[tex]\[ P_{100} = 3070 \][/tex]
