# How do you find r and a1 for the geometric sequence: a3 = 5, a8 = 1/625?

Jan 25, 2016

$r = \frac{1}{5}$
${a}_{1} = 125$

#### Explanation:

For a geometric series:
$\textcolor{w h i t e}{\text{XXX}} {a}_{m} = {a}_{n} \cdot {r}^{n - m}$

We will use this general formula in two forms
$\textcolor{w h i t e}{\text{XXX}} {a}_{8} = {a}_{3} \cdot {r}^{5}$
and
$\textcolor{w h i t e}{\text{XXX}} {a}_{1} = {a}_{3} \cdot {r}^{- 2}$

We are told ${a}_{8} = \frac{1}{625}$ and ${a}_{3} = 5$
Therefore
$\textcolor{w h i t e}{\text{XXX}} \frac{1}{625} = 5 \cdot {r}^{5}$

$\textcolor{w h i t e}{\text{XXX}} {r}^{5} = \frac{1}{5 \cdot 625} = \frac{1}{5 \cdot 5 \cdot 4} = \frac{1}{{5}^{5}}$

$\textcolor{w h i t e}{\text{XXX}} r = \frac{1}{5}$

and since ${a}_{1} = {a}_{3} \cdot {r}^{- 2}$
$\textcolor{w h i t e}{\text{XXX}} {a}_{1} = 5 \cdot {\left(\frac{1}{5}\right)}^{- 2} = 5 \cdot {5}^{2} = 125$