If a, b, c are in G.P., prove that the following are also in G.P. :
a2 + b2, ab + bc, b2 + c2
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.