Question

Which explicit formula can be used to find the number of monkeys in the nth generation?

Answer 1

Solution:

Given that the first generation has 4 monkeys, the monkey population doubles after generation.

This implies that in the second generation, there are 8 monkeys and 16 monkeys in the third generation.

This gives a geometric sequence, whose explicit formula is expressed as

$\begin{gathered} a_n=ar^{(n-1)} \\ where \\ a_n\Rightarrow nth\text{ term} \\ a\Rightarrow first\text{ term} \\ r\Rightarrow common\text{ ratio} \end{gathered}$

The common ratio is evaluated as

$\begin{gathered} r=\frac{second\text{ term}}{first\text{ term}}\text{ or}\frac{third\text{ term}}{secon\text{ term}} \\ thus, \\ r=\frac{8}{4}=2 \end{gathered}$

where

$\begin{gathered} a=4 \\ r=2 \end{gathered}$

By substitution, we have

$a_n=4\times(2)^{(n-1)}$

Hence, the explicit formula to find the number of monkeys in the nth generation is

$a_n=4\cdot2^{(n-1)}$

The correct option is