QuestionFind the explicit formula for the number of rabbits in generation $n$ if there are 108 rabbits in generation 3: $a_{n}=6 \cdot 3^{(n-1)}$ .

Studdy Solution

STEP 1

Assumptions1. The number of rabbits in each generation forms a geometric sequence. . The first term of the sequence is6 (the number of rabbits in the first generation).
3. The common ratio of the sequence is3 (the number of rabbits triples in each generation).

STEP 2

The explicit formula for a geometric sequence is given by $a_{n} = a_{1} \cdot r^{(n-1)}$ where- $a_{n}$ is the nth term of the sequence, - $a_{1}$ is the first term of the sequence, - $r$ is the common ratio of the sequence, and- $n$ is the term number.

STEP 3

We can substitute the given values into the formula. The first term $a_{1}$ is6 (the number of rabbits in the first generation), and the common ratio $r$ is3 (the number of rabbits triples in each generation).
$a_{n} =6 \cdot3^{(n-1)}$ This is the explicit formula that can be used to find the number of rabbits in the nth generation.

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