# How do you determine whether the sequence 1, 1/2, 1/3, 1/4,... is geometric and if it is, what is the common ratio?

Jul 16, 2017

This is not a geometric progression. It is a harmonic one.

#### Explanation:

Examine to see if there's a common ratio:

$\frac{\frac{1}{2}}{1} = \frac{1}{2}$

$\frac{\frac{1}{3}}{\frac{1}{2}} = \frac{2}{3}$

$\frac{\frac{1}{4}}{\frac{1}{3}} = \frac{3}{4}$

The ratio between successive terms is not common, so this is not a geometric sequence.

It is a harmonic sequence - the reciprocals of successive terms being in arithmetic progression.

Jul 16, 2017

$\text{not geometric}$

#### Explanation:

$\text{for the sequence of terms to be geometric there must be a }$
$\text{common ratio (r) between them}$

$\text{where } r = \frac{{a}_{2}}{{a}_{1}} = \frac{{a}_{3}}{{a}_{2}} = \ldots \ldots = \frac{{a}_{n}}{{a}_{n - 1}}$

$\text{here } r = \frac{\frac{1}{2}}{1} = \frac{1}{2} \ne \frac{\frac{1}{3}}{\frac{1}{2}} = \frac{2}{3}$

$\text{sequence is therefore not geometric}$