a, b, c are in G.PThereforeb2 = ac … (1)We have to prove a2 + b2, ab + bc, b2 + c2 are in GP orwe 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) = RHSHence a2 + b2, ab + bc, b2 + c2 are in GP.

Q. 10 C5.0( 1 Vote )

If a, b, c are in

Class 11th RD Sharma - Mathematics 20. Geometric Progressions

Answer :

a, b, c are in G.P

Therefore

b² = ac … (1)

We have to prove a² + b^2, 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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