If a, b, c are in G.P., prove that : (a2 – b2), (b2 – c2), (c2 �

1.	If a, b, c are in G.P., prove that : (a2 – b2), (b2 – c2), (c2 – d2) are in G.P.
Answer» a, b, c, d are in G.P. Therefore, bc = ad … (1) b² = ac … (2) c² = bd … (3) To prove: (a² – b²), (b² – c²), (c² – d²) are in G.P, we need to prove that: (a² – b²) (c² – d²) = (b² – c²)² {deduced using GM relation} ∴ RHS = (b² – c²)² = b⁴ + c⁴ – 2b²c² = a²c² + b²d² – a²d² – b²c² {using equation 2 and 3} = c²(a² – b²) – d²(a² – b²) = (a² – b²) (c² – d²) = LHS ∴ (a² – b²), (b² – c²), (c² – d²) are in G.P Hence proved.

If a, b, c are in G.P., prove that : (a2 – b2), (b2 – c2), (c2 – d2) are in G.P.

Answer»

a, b, c, d are in G.P. Therefore,

bc = ad … (1)

b² = ac … (2)

c² = bd … (3)

To prove: (a² – b²), (b² – c²), (c² – d²) are in G.P, we need to prove that:

(a² – b²) (c² – d²) = (b² – c²)² {deduced using GM relation}

∴ RHS = (b² – c²)²

= b⁴ + c⁴ – 2b²c²

= a²c² + b²d² – a²d² – b²c² {using equation 2 and 3}

= c²(a² – b²) – d²(a² – b²)

= (a² – b²) (c² – d²) = LHS

∴ (a² – b²), (b² – c²), (c² – d²) are in G.P

Hence proved.