# How do you determine whether the sequence 1/5, 2/7, 3/9, 4/11,... is geometric and if it is, what is the common ratio?

Mar 5, 2017

This is not a geometric sequence, since there is no common ratio.

#### Explanation:

If it was a geometric sequence, it would show a common ratio, but this sequence does not. All you need do is test a few consecutive terms to see this:

$\frac{3}{9} \div \frac{2}{7} = \frac{3}{9} \times \frac{7}{2} = \frac{21}{18} = \frac{7}{6}$

$\frac{4}{11} \div \frac{3}{9} = \frac{4}{11} \times \frac{9}{3} = \frac{36}{33} = \frac{12}{11}$

$\frac{5}{13} \div \frac{4}{11} = \frac{5}{13} \times \frac{11}{4} = \frac{55}{52}$

These three ratios are not equal. In fact, the ratios are getting smaller (approaching unity).