# What's the common ratio in 5/12, 1/3, 4/15, 16/75, 64/375?

Dec 3, 2015

$\frac{4}{5}$

#### Explanation:

Divide any number in this sequence by the preceding number and you get $\frac{4}{5}$. That's the common ratio.

For example: $\frac{1}{3} \div \frac{5}{12} = \frac{1}{3} \cdot \frac{12}{5} = \frac{4}{5}$,

$\frac{4}{15} \div \frac{1}{3} = \frac{4}{15} \cdot 3 = \frac{12}{15} = \frac{4}{5}$,

$\frac{16}{75} \div \frac{4}{15} = \frac{16}{75} \cdot \frac{15}{4} = \frac{4}{5}$, and

$\frac{64}{375} \div \frac{16}{75} = \frac{64}{375} \cdot \frac{75}{16} = \frac{4}{5}$.

A cool thing related to this is the corresponding geometric series (infinite "sum") $\frac{5}{12} + \frac{1}{3} + \frac{4}{15} + \frac{16}{75} + \frac{64}{375} + \cdots$. If $a$ is the first term of such a series, and $r$ is the common ratio, with $| r | < 1$, the series converges to $\frac{a}{1 - r}$.

For this example, $a = \frac{5}{12}$ and $r = \frac{4}{5}$ so that the series converges to $\frac{\frac{5}{12}}{1 - \frac{4}{5}} = \frac{\frac{5}{12}}{\frac{1}{5}} = \frac{5}{12} \cdot 5 = \frac{25}{12}$ and we write

$\frac{5}{12} + \frac{1}{3} + \frac{4}{15} + \frac{16}{75} + \frac{64}{375} + \cdots = \frac{25}{12}$.