QuestionFind the formula for rabbit population in the $n$ th generation if it starts with 3 and multiplies by 6 each generation. A. $a_{n}=6 \cdot 3^{(n-1)}$ B. $a_{n}=3 \cdot\left(\frac{1}{6}\right)^{(n-1)}$ C. $a_{n}=6 \cdot\left(\frac{1}{3}\right)^{(n-1)}$ D. $a_{n}=3 \cdot 6^{(n-1)}$

Studdy Solution

STEP 1

Assumptions1. The initial population of rabbits is3 (in the1st generation). . The rabbit population multiplies by6 in every subsequent generation.

STEP 2

We are looking for an explicit formula to find the number of rabbits in the $n$ th generation. This is a geometric sequence problem, where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
The general form of a geometric sequence is $a_{n} = a_{1} \cdot r^{(n-1)}$ , where $a_{n}$ is the $n$ th term, $a_{1}$ is the first term, $r$ is the common ratio, and $n$ is the term number.

STEP 3

In our case, the first term $a_{1}$ is3 (the initial population of rabbits), and the common ratio $r$ is6 (the population multiplies by6 in every generation). So, we can write the formula as $a_{n} =3 \cdot6^{(n-1)}$

STEP 4

Now, let's compare our formula with the given options. We can see that our formula matches with option D.
So, the explicit formula to find the number of rabbits in the $n$ th generation is $a_{n} =3 \cdot6^{(n-1)}$ .

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