Question

A population of beetles is growing to a linear growth model with time interval of 1 week. At the start there are 30 beetles, and in the 8th week there are 670 beetles.A) Write a recursive formula for the beetle population.B) Write an explicit equation for the beetle population.C) How many beetles will there be in week 62?D) In what week will the beetle population reach 1000 bottles?

Answer 1

The question can be solved using an arithmetic sequence model.

The general formula for an arithmetic sequence is given to be:

$a_n=a_1+(n-1)d$

where

$\begin{gathered} a_n=\text{ nth term of the sequence} \\ a_1=\text{ first term} \\ d=\text{ common difference} \\ n=\text{ number/position of nth term} \end{gathered}$

We can get the common difference by applying the information given into the formula.

GIVEN:

$\begin{gathered} a_0=30\text{ (Initial population)} \\ a_8=670 \end{gathered}$

We can model the formula for the 8th term such that:

$a_8=a_1+(8-1)d$

We have the value for the first term to be:

$a_1=a_0+d$

Substituting the values given, we have:

$\begin{gathered} 670=(a_0+d)+7d \\ 670=30+d+7d \\ 670=30+8d \\ 8d=670-30 \\ 8d=640 \\ d=\frac{640}{8} \\ d=80 \end{gathered}$

QUESTION A:

The general recursive formula is given to be:

$a_n=a_{n-1}+d$

Therefore, for this sequence, the recursive formula is:

$a_n=a_{n-1}+80$

QUESTION B:

The general explicit formula for an arithmetic sequence is given to be:

$a_n=a_1+(n-1)d$

Therefore, for this sequence, the explicit formula is given to be:

$\begin{gathered} a_n=a_0+d+(n-1)d \\ a_n=a_0+d+dn-d \\ a_n=a_0+dn \\ \text{Substituting values of }a_0\text{ and d, we have:} \\ a_n=30+80n \end{gathered}$

QUESTION C:

In week 62, we will take n = 62:

$\begin{gathered} a_{62}=30+80(62) \\ a_{62}=4990 \end{gathered}$

There will be 4990 beetles in week 62.

QUESTION D:

In this problem, we will make the following substitution:

$a_n=1000$

Therefore, we can substitute into the explicit formula and solve for n as shown:

$\begin{gathered} 1000=30+80n \\ 80n=1000-30 \\ 80n=970 \\ n=\frac{970}{80} \\ n=12.125 \end{gathered}$

Therefore, in the 13th week, since we can't use a decimal for the weeks, the beetle population will reach 1000 beetles.