Answer :

a, b, c, d are in G.P.


Therefore,


bc = ad … (1)


b2 = ac … (2)


c2 = bd … (3)


If somehow we use RHS and Make it equal to LHS, our job will be done.


we can manipulate the RHS of the given equation as –


Note: Here we are manipulating RHS because working with a simpler algebraic equation is easier and this time RHS is looking simpler.


RHS = (a + b)2 + 2(b + c)2 + (c + d)2


RHS = a2 + b2 + 2ab + 2(c2 + b2 + 2cb) + c2 + d2 + 2cd


RHS = a2 + b2 + c2 + d2 + 2ab + 2(c2 + b2 + 2cb) + 2cd


Put c2 = bd and b2 = ac, we get –


RHS = a2 + b2 + c2 + d2 + 2(ab + ad + ac + cb + cd)


You can visualize the above expression by making separate terms for (a + b + c)2 + d2 + 2d(a + b + c) = {(a + b + c) + d}2


RHS = (a + b + c + d)2 = LHS


Hence Proved.


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