# The sum of three consecutive odd integers is -87. What are the integers?

May 15, 2016

$\left\{- 31 , - 29 , - 27\right\}$

#### Explanation:

Any odd integer may be expressed as $2 n + 1$ for some integer $n$. As we are looking for three consecutive odd integers, we will represent the least as $2 n + 1$, and the next two as $2 n + 3$, and $2 n + 5$. With that, we have

$\left(2 n + 1\right) + \left(2 n + 3\right) + \left(2 n + 5\right) = - 87$

$\implies 6 n + 9 = - 87$

$\implies 6 n = - 96$

$\implies n = - 16$

Then, the three odd integers are

$\left\{2 \left(- 16\right) + 1 , 2 \left(- 16\right) + 3 , 2 \left(- 16\right) + 5\right\}$

$= \left\{- 31 , - 29 , - 27\right\}$

May 15, 2016

$- 31 , - 29 , - 27$

#### Explanation:

Alternatively, suppose the second consecutive odd integer is $n$.

Then the first and third are $\left(n - 2\right)$ and $\left(n + 2\right)$.

So:

$- 87 = \left(n - 2\right) + n + \left(n + 2\right) = 3 n$

Divide both ends by $3$ to get:

$- 29 = n$

So the three consecutive odd integers are:

$- 31 , - 29 , - 27$