Question

A population of beetles are growing according to a linear growth model. The initial population (week 0) is P0=4 , and the population after 6 weeks is P6=34 .

Answer 1

ANSWER

$\begin{gathered} P_n=4+5n \\ n=23 \end{gathered}$

EXPLANATION

Given:

$\begin{gathered} P_0=4 \\ P_6=34 \end{gathered}$

Desired Outcome:

1. Explicit formula (Pn)

2. Number of weeks the population of the beetle will reach 119

Note: This series is Arithmetic Progression with first term of P0 = 4

Now, let's determine the COMMON DIFFERENCE

$\begin{gathered} common\text{ }difference=\frac{34-4}{6} \\ =\frac{30}{6} \\ =5 \end{gathered}$

Therefore, every week 5 beetles are added.

The explicit formula becomes:

$P_n=4+5n$

Determine the number of weeks the population of the beetles will reach 119

$\begin{gathered} P_n=119=4+5n \\ 119-4=5n \\ 115=5n \\ n=\frac{115}{5} \\ n=23 \end{gathered}$

Hence, the number of weeks the population of the beetles will reach 119 is 23 weeks