Question

How do you find three consecutive odd integers?

Answer 1

Three consecutive odd integers are always of the form `(2k+1)`, `(2k+3)` and `(2k+5)` for some integer `k`.

Explanation:

Every odd integer can be expressed in the form `2k+1` for some integer `k`.

Given an odd integer `2k+1`, the next odd integer is `(2k+1)+2 = 2k+3` and the next after that is `2k+5`.

If you want to find three consecutive odd integers to satisfy some property then you can substitute these expressions into the condition and solve for `k`.

For example, to find three consecutive odd integers whose sum is `189` you can write:

`189 = (2k+1) + (2k+3) + (2k+5) = 6k+9`

Subtract `9` from both sides to get:

`180 = 6k`

Divide both sides by `6` to get `k = 30`.

Then our three odd integers are

`(2k + 1) = 61`, `(2k+3) = 63` and `(2k+5) = 65`