If a, b, c are in G.P., prove that the following are also in G.P. : a

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

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

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.