If a, b, c are in G.P, then show that a2 + b2, ab + bc, b2 + c2 are al

1.	If a, b, c are in G.P, then show that a2 + b2, ab + bc, b2 + c2 are also in G.P.
Answer» Given that, a, b, c are in G.P ⇒ b² = ac ⇒ b² − ac = 0 ⇒ (b² − ac)² = 0 ⇒ b⁴ + a²c² − 2b²ac = 0 ⇒ a²b² + b²c² + 2b²ac = a²b² + b²c² + a²c² + b⁴ (Adding a²b² + b²c² both sides) ⇒ (ab + bc)² = (a²+ b²)(b²+ c²) ⇒ a²+ b², ab + bc, b²+ c² are in G.P. Hence Proved

If a, b, c are in G.P, then show that a2 + b2, ab + bc, b2 + c2 are also in G.P.

Answer»

Given that, a, b, c are in G.P

⇒ b² = ac

⇒ b² − ac = 0

⇒ (b² − ac)² = 0

⇒ b⁴ + a²c² − 2b²ac = 0

⇒ a²b² + b²c² + 2b²ac = a²b² + b²c² + a²c² + b⁴

(Adding a²b² + b²c² both sides)

⇒ (ab + bc)² = (a²+ b²)(b²+ c²)

⇒ a²+ b², ab + bc, b²+ c² are in G.P.

Hence Proved