A population of beetles are growing according to a linear growth model. The initial population (week 0) is  P0=4, and the population after 4 weeks is  P4=52
 .

(a) Find an explicit formula for the beetle population after  n
  weeks.


The beetle population is modeled using a linear growth equation, which generally takes the form:

\[
P(n) = P_0 + rn
\]

where:
- \( P(n) \) is the population after \( n \) weeks,
- \( P_0 \) is the initial population,
- \( r \) is the constant rate of growth per week,
- \( n \) is the number of weeks.

### Step 1: Find the rate of growth (\( r \))

Given the following information:
- Initial population at \( n = 0 \), \( P_0 = 4 \),
- Population at \( n = 4 \), \( P_4 = 52 \).

We can set up the equation for \( n = 4 \):

\[
P(4) = P_0 + 4r
\]

Substitute the known values:

\[
52 = 4 + 4r
\]

Now, solve for \( r \):

\[
52 - 4 = 4r
\]
\[
48 = 4r
\]
\[
r = \frac{48}{4} = 12
\]

So, the rate of growth is \( r = 12 \) beetles per week.

### Step 2: Write the explicit formula

Now that we know the rate of growth, the explicit formula for the population after \( n \) weeks is:

\[
P(n) = 4 + 12n
\]

### Final Answer:
The formula for the beetle population after \( n \) weeks is:

\[
P(n) = 12n + 4
\]

Would you like further details or have any questions?

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Here are 5 related questions to consider:
1. What will the beetle population be after 6 weeks?
2. How many weeks will it take for the population to reach 100 beetles?
3. If the growth model changes from linear to exponential, how would the formula change?
4. What is the significance of the constant 4 in the equation \( P(n) = 12n + 4 \)?
5. Can you create a graph that represents the population growth over time?

**Tip**: Linear growth models assume a constant rate of change, so the population increases by the same amount each period.

A population of beetles are growing according to a linear growth model. The initial population (week 0) is P0=4, and the population after 4 weeks is P4=52. (a) Find an explicit formula for the beetle population after n weeks.

Discover how to find the explicit formula for beetle population growth using a linear model. Given an initial population of 4 and a population of 52 after 4 weeks, this step-by-step guide walks through solving for the growth rate and formulating the equation. Ideal for students in Grades 7-9 studying linear growth and algebra.

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