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Given: a, b and c ar in AP
As we know, if a, b and c are in AP then 2b = a + c------ (1)
We have to prove that ( bc - a2 ), ( ca - b2 ), ( ab - c2 ) are in AP.
For this we should prove : 2( ca - b2 ) = ( bc - a2 )+ ( ab - c2 )
Let us consider ( bc - a2 )+ ( ab - c2 )
= ab + bc - (a2+c2)
= b( a + c ) - [ (a + c)2 - 2ac] [ ∵ a2+b2 = (a + b)2 - 2ab ]
= b(2b) - [ ( 2b )2 -2ac] [ ∵ From (1) ]
= 2b2 - 4b2 + 2ac
= 2ac - 2b2
= 2(ca - b2)
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