Question

The common ratio of the geometric sequence -14,-84,-504

Answer 1

A common ratio is basically a ratio of next sequence and the sequence you want

To find a common ratio of geometric sequence, we usually define 'r' as common ratio.

$\displaystyle \large \tt{r = \frac{next \: \: sequence}{the \: \: sequence \: \: you \: \: want} }$

For example, I want to focus on -14 and the next sequence would be -84.

Hence,

$\displaystyle \large{r = \frac{ - 84}{ - 14} }$

Thus, r is 6 because both numerator and denominator are negative. negative divides negative = positive.

Or you want to choose -84, then the next sequence would be -504.

$\displaystyle \large{r = \frac{ - 504}{ - 84} }$

Then r would still be 6. Since both ways have r as 6 and this proves that the sequence is geometric.

Hence, the common ratio is 6.