How do you tell whether the sequence -6, -2, -2/3, -2/9, -2/27 is geometric?

May 3, 2017

see explanation.

Explanation:

$\text{the terms of a geometric sequence are}$

$a , a r , a {r}^{2} , a {r}^{3} , \ldots . , a {r}^{n - 1}$

$\text{where r is the common ratio and}$

$r = \frac{{a}_{2}}{{a}_{1}} = \frac{{a}_{3}}{{a}_{2}} = \ldots . = \frac{{a}_{n}}{{a}_{n - 1}}$

$\text{ check r is common to the terms in this sequence}$

$\frac{{a}_{2}}{{a}_{1}} = \frac{- 2}{- 6} = \frac{1}{3}$

$\frac{{a}_{3}}{{a}_{2}} = \frac{- \frac{2}{3}}{- 2} = \frac{1}{3}$

$\frac{{a}_{4}}{{a}_{3}} = \frac{- \frac{2}{9}}{- \frac{2}{3}} = \frac{1}{3}$

$\Rightarrow \text{ sequence is geometric}$