Question

What is the 7th term in the geometric sequence in which a10 is 9 and a13 is -72?

Answer 1

Recall, a geometric series has this form with constants A for n=0, and k, the ratio of two consecutive terms n/(n+1)
$a_n =A*k^n$

you are given two values, for n=13 and n=10, put both into the equation

$9=A*k^{10}$
$-72=A*k^{13}$


Two equations with two unknown values A and k, you can solve for those.

Solving both for A,
$A= \frac{9}{k^{10}} = \frac{-72}{k^{13}}$


$k=-2$
$A= \frac{9}{1024}$

The general equation for the series is then,
$a_n= \frac{9}{1024}(-2)^n$

use n=7