# A geometric sequence has first term 54 and 4th term 2. What is the common ratio?

Nov 8, 2015

$r = \frac{1}{3}$

#### Explanation:

${A}_{n} = {A}_{1} {r}^{n - 1}$

${A}_{1} = 54$

${A}_{4} = 2 = {A}_{1} {r}^{4 - 1}$

$\implies 2 = 54 {r}^{3}$

$\implies \frac{2}{54} = {r}^{3}$
$\implies \frac{1}{27} = {r}^{3}$

$\implies {\left(\frac{1}{3}\right)}^{3} = {r}^{3}$

$\implies r = \frac{1}{3}$