Sure, let's solve the problem step by step.
### Step 1: Calculate the growth rate
First, we need to determine the weekly growth rate of the beetle population. We know:
- At week 0, the population [tex]\( P_0 = 5 \)[/tex]
- At week 8, the population [tex]\( P_8 = 53 \)[/tex]
The growth over these 8 weeks is given by the difference between [tex]\( P_8 \)[/tex] and [tex]\( P_0 \)[/tex]:
[tex]\[
P_8 - P_0 = 53 - 5 = 48
\][/tex]
Since this growth occurred over 8 weeks, the growth rate per week is:
[tex]\[
\text{Growth rate} = \frac{P_8 - P_0}{8} = \frac{48}{8} = 6
\][/tex]
### Step 2: Construct the explicit formula
We want to find the population after [tex]\( n \)[/tex] weeks, [tex]\( P_n \)[/tex]. Given the linear growth model, the population increases by a constant amount each week, which we've calculated to be 6.
The general formula for the population after [tex]\( n \)[/tex] weeks can be written as:
[tex]\[
P_n = P_0 + (\text{growth rate}) \times n
\][/tex]
Substitute [tex]\( P_0 \)[/tex] and the growth rate:
[tex]\[
P_n = 5 + 6n
\][/tex]
### Step 3: Calculate the number of weeks to reach a population of 131
Now we need to determine after how many weeks the population will reach 131. We set [tex]\( P_n = 131 \)[/tex] and solve for [tex]\( n \)[/tex]:
[tex]\[
131 = 5 + 6n
\][/tex]
Subtract 5 from both sides:
[tex]\[
126 = 6n
\][/tex]
Divide by 6:
[tex]\[
n = \frac{126}{6} = 21
\][/tex]
### Summary
1. The explicit formula for the beetle population after [tex]\( n \)[/tex] weeks is:
[tex]\[
P_n = 5 + 6n
\][/tex]
2. The beetle population will reach 131 after:
[tex]\[
21 \text{ weeks}
\][/tex]
