The term independent of x in the simplified form of the expression _ k=0 ³⁰ (x^2 + 1 x )^ 30-k (x^2 - 1 x )^k is

Step-by-step solution

The given expression is S = _ k=0 ³⁰ (x^2 + 1 x )^ 30-k (x^2 - 1 x )^k . This series is a geometric progression with the first term a = (x^2 + 1 x )³⁰ , common ratio r = x^2 - 1 x x^2 + 1 x , and 31 terms. The sum of the G.P. is: S = (x^2 + 1 x )³¹ - (x^2 - 1 x )³¹ (x^2 + 1 x ) - (x^2 - 1 x ) S = (x^2 + 1 x )³¹ - (x^2 - 1 x )³¹ 2 x = x 2 [ (x^2 + 1 x )³¹ - (x^2 - 1 x )³¹ ] We know that (A+B)^n - (A-B)^n = 2 [ ^ n C₁ A^ n-1 B^1 + ^ n C₃ A^ n-3 B^3 + ] . Here, A = x^2 , B = 1 x , and n = 31 . The general term in the