If S₁,

Sum of GP for n terms S₁ =

= …(1)

Sum of GP for 2n terms S₂ =

= …(2)

Sum of GP for 3n terms S₃ = …(3)

=

Let

∴ 1, 2 and 3 becomes

S_{1 =} K(rⁿ - 1)

S_{2 =} K(r²ⁿ - 1)

S_{3 =} K(r³ⁿ - 1)

∴ S₁² + S²_{2 =} k²(rⁿ - 1)² + k²(r²ⁿ - 1)²

⁼ k²(r²ⁿ + 1 - 2.rⁿ + r⁴ⁿ + 1 - 2.r²ⁿ)

= k²(r⁴ⁿ - r²ⁿ - 2rⁿ + 2)

L.H.S = k²(r⁴ⁿ - r²ⁿ - 2rⁿ + 2)

S₁(S₂ + S₃) = K(rⁿ - 1)[ (K(r²ⁿ - 1) + K(r³ⁿ - 1))]

= K²(rⁿ - 1)[r²ⁿ + r³ⁿ - 2]

= k²(r⁴ⁿ - r²ⁿ - 2rⁿ + 2)

Hence, Proved.