# The second and fifth term of a geometric series are 750 and -6 respectively. Find the common ratio of and the first term of the series?

Mar 1, 2017

$r = - \frac{1}{5} , {a}_{1} = - 3750$

#### Explanation:

The $\textcolor{b l u e}{\text{nth term of a geometric sequence}}$ is.

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{{a}_{n} = a {r}^{n - 1}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$
where a is the first term and r, the common ratio.

$\Rightarrow \text{second term } = a {r}^{1} = 750 \to \left(1\right)$

$\Rightarrow \text{fifth term } = a {r}^{4} = - 6 \to \left(2\right)$

To find r, divide ( 2) by ( 1)

$\Rightarrow \frac{\cancel{a} {r}^{4}}{\cancel{a} r} = \frac{- 6}{750}$

$\Rightarrow {r}^{3} = - \frac{1}{125} \Rightarrow r = - \frac{1}{5}$

Substitute this value into ( 1) to find a

$\Rightarrow a \times - \frac{1}{5} = 750$

$\Rightarrow a = \frac{750}{- \frac{1}{5}} = - 3750$