A.P !

Number Theory Level 3

If $x$ , $z$ and $y$ are in an arithmetic progression (in that order), where $x$ , $y$ and $z$ are distinct real numbers, evaluate $\dfrac{x}{x-z} + \dfrac{z}{y-z}$ .

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Chew-Seong Cheong
Dec 25, 2017

Since $x$ , $z$ and $y$ are in an arithmetic progression, let $x = z-d$ and (y = z+d), where $d$ is the common difference. Then we have:

$\begin{aligned} \frac x{x-z} + \frac z{y-z} & = \frac {z-d}{-d} + \frac zd = - \frac zd + 1 + \frac zd = \boxed{1} \end{aligned}$

A.P !

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