If $a, b, c$ are in arithmetic progression, then the value of $(a + 2 b - c) (2 b + c - a) (a + 2 b + c)$

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Solution

Finding the value of $left parenthesis a space plus space 2 b space minus space c right parenthesis left parenthesis 2 b space plus space c space minus space a right parenthesis left parenthesis a space plus space 2 b space plus space c right parenthesis$ :
Given $a, b, c$ are in $A . P$
So $2 b = a + c$
and $(a + 2 b - c) (2 b + c - a) (a + 2 b + c) . . . . (i)$
we get $(a + a + c - c) (a + c + c - a) (a + a + c + c)$
$= (2 a) (2 c) (2 a + 2 c) = 8 a c (a + c) = 8 a c (2 b) = 16 a b c$
Therefore $(a + 2 b - c) (a + 2 b + c) (2 b + c - a) = 16 a b c$
Hence, the value of $(a + 2 b - c) (2 b + c - a) (a + 2 b + c)$ is $16 a b c$

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