To solve the given population growth problem step-by-step, let's follow the outlined tasks methodically:
1. Calculate [tex]\( P_1 \)[/tex]:
Given the initial population [tex]\( P_0 = 80 \)[/tex] and the recursive rule [tex]\( P_n = P_{n-1} + 95 \)[/tex]:
[tex]\[
P_1 = P_0 + 95 = 80 + 95 = 175
\][/tex]
So, [tex]\( P_1 = 175 \)[/tex].
2. Calculate [tex]\( P_2 \)[/tex]:
Using the previously found [tex]\( P_1 \)[/tex] and the same recursive rule:
[tex]\[
P_2 = P_1 + 95 = 175 + 95 = 270
\][/tex]
So, [tex]\( P_2 = 270 \)[/tex].
3. Find an explicit formula for the population [tex]\( P_n \)[/tex]:
The recursive rule [tex]\( P_n = P_{n-1} + 95 \)[/tex] suggests that each subsequent term increases by 95. To derive an explicit formula, observe that each step adds an additional 95. Starting from [tex]\( P_0 \)[/tex]:
- [tex]\( P_1 = P_0 + 95 \)[/tex]
- [tex]\( P_2 = P_0 + 2 \times 95 \)[/tex]
- [tex]\( P_3 = P_0 + 3 \times 95 \)[/tex]
Generalizing this:
[tex]\[
P_n = P_0 + n \times 95
\][/tex]
Substituting [tex]\( P_0 = 80 \)[/tex]:
[tex]\[
P_n = 80 + n \times 95
\][/tex]
So, the explicit formula is [tex]\( P_n = 80 + n \times 95 \)[/tex].
4. Calculate [tex]\( P_{100} \)[/tex] using the explicit formula:
Now, substitute [tex]\( n = 100 \)[/tex] into the formula [tex]\( P_n = 80 + n \times 95 \)[/tex]:
[tex]\[
P_{100} = 80 + 100 \times 95 = 80 + 9500 = 9580
\][/tex]
So, [tex]\( P_{100} = 9580 \)[/tex].
Summary:
[tex]\[
P_1 = 175
\][/tex]
[tex]\[
P_2 = 270
\][/tex]
[tex]\[
P_n = 80 + n \times 95
\][/tex]
[tex]\[
P_{100} = 9580
\][/tex]
