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Sum of N Terms of an AP

If pth,qth ...

Question

If $p^{t h}, q^{t h}$ and $r^{t h}$ terms of an A.P. are $a, b, c$ respectively, then find the value of:
$(a - b) r + (b - c) p + (c - a) q$

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Solution

Let us consider
First term $= A$
Common difference $= D$
then $a = A + (p - 1) D$
$b = A + (q - 1) D$
$c = A + (r - 1) D$
Given
$(a - b) r + (b - c) p + (c - a) q$
$= [A + (p - 1) D - A - (q - 1) D] r + [A + (q - 1) D - A - (r - 1) D] p + [A + (r - 1) D - A - (p - 1) D] q$
$= (p - q) D r + (q - r) D p + (r - p) D q$
$= D [p r - q r + q p - r p + r q - p q]$
$= D (0) = 0$

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If $p^{t h}$ , $q^{t h}$ and $r^{t h}$ terms of an A.P. are a, b, c respectively, then show that: $a (q - r) + b (r - p) + c (p - q) = 0$

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