Which term of the G.P.: $2, 8, 32, \dots$ is $131072$? Write the explicit formula as well as the recursive formula for the $n^\text{th}$ term.

Given GP: 2, 8, 32, ...

First term, $a = 2$, and common ratio, $r = \frac{8}{2} = 4$.

$n^{\text{th}}$ term formula of a GP: $a_n = ar^{n-1}$

So, $$a_n = 2(4)^{n-1}$$

We need: $$2(4)^{n-1} = 131072$$

$$(4)^{n-1} = 65536$$

Since, $$65536 = 4^8$$

therefore, $$n - 1 = 8$$

$$n = 9$$

So, the $9^{\text{th}}$ term of G.P. is $131072$.

Recursive formula, $a_{1} = 2$, $a_{n} = 4a_{n-1}$ for $n \geq 2$.