If $p, q$ and $r$ in $A P$ , then prove that $(p + 2 q - r) (2 q + r - p) (r + p - q) = 4 p q r$ .

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Solution

$p, q$ and $r$ are in $A P$
i.e., $2 q = p + r$
$(p + 2 q - r) (2 q + r - p) (r + p - q)$
$= (p + p + r - r) (p + r + r - p) (2 q - q)$
$= (2 p) (2 r) (q)$
$= 4 p q r$
Hence proved

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