Question

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

Answer 1

`4/5`

Explanation:

Divide any number in this sequence by the preceding number and you get `4/5`. That's the common ratio.

For example: `1/3 div 5/12=1/3 * 12/5=4/5`,

`4/15 div 1/3=4/15 * 3=12/15=4/5`,

`16/75 div 4/15=16/75 * 15/4=4/5`, and

`64/375 div 16/75 = 64/375 * 75/16 = 4/5`.

A cool thing related to this is the corresponding geometric series (infinite "sum") `5/12+1/3+4/15+16/75+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 `a/(1-r)`.

For this example, `a=5/12` and `r=4/5` so that the series converges to `(5/12)/(1-4/5)=(5/12)/(1/5)=5/12 * 5=25/12` and we write

`5/12+1/3+4/15+16/75+64/375+cdots=25/12`.