Question

How to find the sum of the harmonic sequence below? `1/3` + `1/6` + `1/9` + `1/12`

Answer 1

The sum is `{25}/{36}`.

Explanation:

Put all fractions over a common denominator. That denominator must be a multiple of the given ones since the fractions `1/3, 1/6, 1/9, 1/{12}` are all in lowest terms. The smallest whole number that is a multiple of all four denominator, called the least common multiple, is `36`, so we choose that common denominator.

`1/3=1/3×{12}/{12}={12}/{36}`

`1/6=1/6×6/6=6/{36}`

`1/9=1/9×4/4=4/{36}`

`1/{12}=1/{12}×3/3=3/{36}`

Then just add the numerators, and see if the resulting fraction can be reduced to lower terms:

`{12}/{36}+6/{36}+4/{36}+3/{36}={12+6+4+3}/{36}={25}/{36}`.

The sum cannot be reduced to lower terms because `25` and `36` habe no common factors greater than one.