# The average of five consecutive odd integers is -21. What is the least of these integers?

Feb 27, 2018

$- 25$

#### Explanation:

Take $x$. This is the smallest integer. Since these are consecutive odd integers, the second must be $2$ greater than the first. The third number must be $2$ greater than the second. And so forth.

For example, $1 , 3 , 5 , 7 , \mathmr{and} 9$ are five consecutive odd integers, and they are all two more than the last. So, our five numbers are

$x , x + 2 , \left(x + 2\right) + 2 , \left(\left(x + 2\right) + 2\right) + 2 , \mathmr{and} \left(\left(\left(x + 2\right) + 2\right) + 2\right) + 2$

which means

$x , x + 2 , x + 4 , x + 6 , \mathmr{and} x + 8$

According to the question, their average is $- 21$. So,

$\frac{x + \left(x + 2\right) + \left(x + 4\right) + \left(x + 6\right) + \left(x + 8\right)}{5} = - 21$

Therefore, by simplifying,

$\frac{5 x + 20}{5} = - 21$

So

$5 x + 20 = - 105$

Then

$5 x = - 125$

and

$x = - 25$

Shortcut: Since these are odd integers that are consecutive, you can take $- 21$ as the middle number, $- 23$ as the second, $- 19$ to even out the $- 23$ and maintain the average of $- 21$, then $- 25$ as the first, then $- 17$ as the last. This is a little hard to explain but makes sense if you really think about it.

Feb 27, 2018

$\text{The answer is:} \setminus q \quad \setminus q \quad \setminus q \quad - 25.$

$\text{[And the 5 integers in question are:}$

 \qquad \qquad \qquad \qquad \qquad \quad -25, -23, -21, -19, -17.]

#### Explanation:

$\text{Let the smallest of these odd integers be:} \setminus q \quad \setminus q \quad 2 n - 1.$

$\text{The remaining 4 odd integers are:}$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus \quad \setminus 2 n + 1 , \setminus \quad 2 n + 3 , \setminus \quad 2 n + 5 , \setminus \quad 2 n + 7. \setminus \quad$

$\text{The average of all 5 odd integers is:}$

$\frac{\left(2 n - 1\right) + \left(2 n + 1\right) + \left(2 n + 3\right) + \left(2 n + 5\right) + \left(2 n + 7\right)}{5.}$

$\text{The average of all 5 odd integers is given to be -21. Thus:}$

$\frac{\left(2 n - 1\right) + \left(2 n + 1\right) + \left(2 n + 3\right) + \left(2 n + 5\right) + \left(2 n + 7\right)}{5}$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus \quad \setminus \quad = - 21.$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \frac{5 \left(2 n\right) - 1 + 1 + 3 + 5 + 7}{5} \setminus = \setminus - 21$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \frac{10 n + 15}{5} \setminus = \setminus - 21$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus \frac{\textcolor{red}{\cancel{5}} \left(2 n + 3\right)}{\textcolor{red}{\cancel{5}}} \setminus = \setminus - 21$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus 2 n + 3 \setminus = \setminus - 21$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus \quad \setminus \setminus 2 n \setminus = \setminus - 21 - 3$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus \quad \setminus \setminus 2 n \setminus = \setminus - 24$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus \quad \setminus \setminus \setminus \setminus n \setminus = \setminus - 12.$

$\text{At the start here, we had:}$

$\setminus q \quad \setminus q \quad \setminus \quad \setminus \text{the smallest of these odd integers is:} \setminus q \quad \setminus \quad 2 n - 1.$

$\text{As we found" \quad n \ = -12, \ "we have:}$

$\setminus q \quad \setminus q \quad \setminus \quad \setminus \text{the smallest of these odd integers is:}$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad = \setminus 2 \left(- 12\right) - 1$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad = \setminus - 24 - 1$

$\setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad \setminus q \quad = \setminus - 25.$

 "This is our answer:" \qquad \qquad \qquad -25. \qquad \qquad \qquad \qquad \qquad \qquad !!