# How do you find three consecutive odd integers?

Jun 10, 2015

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

#### Explanation:

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

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

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:

$189 = \left(2 k + 1\right) + \left(2 k + 3\right) + \left(2 k + 5\right) = 6 k + 9$

Subtract $9$ from both sides to get:

$180 = 6 k$

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

Then our three odd integers are

$\left(2 k + 1\right) = 61$, $\left(2 k + 3\right) = 63$ and $\left(2 k + 5\right) = 65$