Question

If S1, S2, S3 be respectively the sums of n, 2n, 3n terms of S1 2 + S2 2=S1 (S2+S3 ).​

Answer 1

Answer:

Step-by-step explanation:

Let the series in GP be a, ar, ar², ar³ and so on.

S₁ = Sₙ = a(rⁿ - 1) / r - 1

S₂ = S₂ₙ = a(r²ⁿ - 1) / r - 1

S₃ = S₃ₙ = a(r³ⁿ - 1) / r - 1

Now,

S₁² = [a² / (r - 1)²] *  [(rⁿ - 1)²]

S₂² = [a²/(r - 1)²] * [(r²ⁿ - 1)²]

L.H.S:

S₁² + S₂² = [a² / (r - 1)²] *  [(rⁿ - 1)²]  +  [a²/(r - 1)²] * [(r²ⁿ - 1)²]

             = [a² / (r - 1)²]  [(rⁿ - 1)² + (r²ⁿ - 1)²]

              = [a² / (r - 1)²] [ r²ⁿ - 2rⁿ + 1 + r⁴ⁿ - 2r²ⁿ + 1]

              = [a² / (r - 1)²] [ r⁴ⁿ - r²ⁿ -  2rⁿ + 2]

R.H.S:

S₁ ( S₂ + S₃) =    [a(rⁿ - 1) / r - 1 ] [ a(r²ⁿ - 1) / r - 1  + a(r³ⁿ - 1) / r - 1]

                   = [a(rⁿ - 1) / r - 1 ] [(a /r - 1)( r²ⁿ + r³ⁿ - 2)]

                   = [a² / (r - 1)²] [ (rⁿ - 1)(r²ⁿ + r³ⁿ - 2)]

                   = [a² / (r - 1)²] [r³ⁿ - r²ⁿ + r⁴ⁿ - r³ⁿ - 2rⁿ + 2]

                   = [a² / (r - 1)²][ r⁴ⁿ - r²ⁿ - 2rⁿ + 2]

=> L.H.S = R.H.S

Hence proved.