Answer :

a, b, c are in G.P


Therefore


b2 = ac … (1)


We have to prove a2 + b2, ab + bc, b2 + c2 are in GP or


we need to prove: (ab + bc)2 = (a2 + b2).(b2 + c2) {using GM}


Take LHS and proceed:


LHS = (ab + bc)2 = a2b2 + 2ab2c + b2c2


b2 = ac


LHS = a2b2 + 2b2(b2) + b2c2


LHS = a2b2 + 2b4 + b2c2


LHS = a2b2 + b4 + a2c2 + b2c2 {again using b2 = ac }


LHS = b2(b2 + a2) + c2(a2 + b2)


LHS = (a2 + b2)(b2 + c2) = RHS


Hence a2 + b2, ab + bc, b2 + c2 are in GP.


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