# If x+14, 13-x, x+8 is an arithmetic sequence, then what is the value of x ?

May 19, 2017

$x = 1$

#### Explanation:

The difference between $13 x - 1$ and $x + 14$ is:

$\left(13 x - 1\right) - \left(x + 14\right) = 12 x - 15$

The difference between $x + 8$ and $13 x - 1$ is:

$\left(x + 8\right) - \left(13 x - 1\right) = - 12 x + 9$

The given terms form an arithmetic sequence if and only if these two differences are equal. That is:

$12 x - 15 = - 12 x + 9$

Add $12 x + 15$ to both sides to get:

$24 x = 24$

Divide both sides by $24$ to get:

$x = 1$

So this is the only solution, yielding the arithmetic sequence:

$15 , 12 , 9$

May 21, 2017

$x = 1$

As this gives a linear equation, there is only one solution.

#### Explanation:

If you know it is an arithmetic sequence, then you know that the common difference, $d$ between any two consecutive terms is the same.

$d = {T}_{3} - {T}_{2} = {T}_{2} - {T}_{1}$

$d = \left(x + 8\right) - \left(13 x - 1\right) = \left(13 x - 1\right) - \left(x + 14\right)$

$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots} x + 8 - 13 x + 1 = 13 x - 1 - x - 14$

$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .} 9 + 15 = 12 x + 12 x$

$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .} 24 = 24 x$

$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .} 1 = x$