SOLUTION: The xth, yth, and zth terms of a sequence are X,Y,Z respectively. Show that if the sequence is arithmetic then X(y-z) + Y(z-x) + Z(x-y)=0.

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Question 1110010: The xth, yth, and zth terms of a sequence are X,Y,Z respectively. Show that if the sequence is arithmetic then X(y-z) + Y(z-x) + Z(x-y)=0.

Found 2 solutions by AnlytcPhil, Edwin McCravy:
Answer by AnlytcPhil(1806)   (Show Source): You can put this solution on YOUR website!

Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
To avoid getting capital and small letters confused I will 
let p=x, q=y, and r=z.  Then the problem is:

The pth, qth, and rth terms of a sequence are X,Y,Z respectively. Show that
if the sequence is arithmetic then X(q-r) + Y(r-p) + Z(p-q)=0.
Let the sequence be arithmetic, with first term a 
and common difference d, then by the formula for nth term, 
we have these three equations:



Solve the first and second equations for d


Solve the first and third equations for d


Since both expressions equal d, they are equal to each other



Cross-multiply:











Edwin
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