Question 1110010
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To avoid getting capital and small letters confused I will 
let p=x, q=y, and r=z.  Then the problem is:
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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.
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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:

{{{system(a + (p-1)d = X,a + (q-1)d = Y,a + (r-1)d = Z)}}}

Solve the first and second equations for d
{{{d=(Y-X)/(q-p)}}}

Solve the first and third equations for d
{{{d=(Z-X)/(r-p)}}}

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

{{{(Y-X)/(q-p)}}}{{{""=""}}}{{{(Z-X)/(r-p)}}}

Cross-multiply:

{{{(Y-X)(r-p)=(Z-X)(q-p)}}}

{{{Yr-Yp-Xr+Xp = Zq-Zp-Xq+Xp}}}

{{{Yr-Yp-Xr = Zq-Zp-Xq}}}

{{{Xq-Xr+Yr-Yp+Zp-Zq=0}}}

{{{X(q-r) + Y(r-p) + Z(p-q) = 0}}}

Edwin</pre>