# How do you find the common ratio for 64,-32,16,-8,4,...?

Sep 21, 2016

Common ratio is $- \frac{1}{2}$

#### Explanation:

In a Geometric series, common ratio is the ratio of a term with respect to its preceding term.

Here we have the series $\left\{64 , - 32 , 16 , - 8 , 4 , \ldots \ldots \ldots \ldots \ldots . .\right\}$

and hence common ratio is $- \frac{32}{64} = \frac{16}{- 32} = - \frac{8}{16} = \frac{4}{- 8} = - \frac{1}{2}$

Sep 21, 2016

Common ratio: $\textcolor{g r e e n}{- \frac{1}{2}}$

#### Explanation:

Let the ratio between successive terms ${a}_{n}$ and ${a}_{n + 1}$ be ${r}_{n} = \frac{{a}_{n + 1}}{{a}_{n}}$

Then for the given terms: $64 , - 32 , 16 , - 8 , 4 , \ldots$
$\textcolor{w h i t e}{\text{XX}} {r}_{1} = \frac{- 32}{64} = - \frac{1}{2}$

$\textcolor{w h i t e}{\text{XX}} {r}_{2} = \frac{16}{- 32} = - \frac{1}{2}$

$\textcolor{w h i t e}{\text{XX}} {r}_{3} = \frac{- 8}{16} = - \frac{1}{2}$

$\textcolor{w h i t e}{\text{XX}} {r}_{4} = \frac{4}{- 8} = - \frac{1}{2}$

assuming this pattern holds for subsequent (unspecified) terms,
we can see that there is a common ration of $\left(- \frac{1}{2}\right)$