The coefficient of x¹⁰⁰ in the simplified form of the expression (1+x)²⁰⁰ - (1+x)¹⁹⁹(1-x) + (1+x)¹⁹⁸(1-x)^2 - + (1-x)²⁰⁰ is :

Step-by-step solution

Let the given sum be S . The given series is a geometric progression with the first term a = (1+x)²⁰⁰ , common ratio r = - 1-x 1+x , and the number of terms n = 201 . The sum of the geometric progression is given by S = a 1 - r^n 1 - r . S = (1+x)²⁰⁰ 1 - (- 1-x 1+x )²⁰¹ 1 - (- 1-x 1+x ) S = (1+x)²⁰⁰ 1 + (1-x)²⁰¹ (1+x)²⁰¹ 1 + 1-x 1+x S = (1+x)²⁰⁰ (1+x)²⁰¹ + (1-x)²⁰¹ (1+x)²⁰¹ 2 1+x S = (1+x)²⁰¹ + (1-x)²⁰¹ 2 Expanding the binomials, the odd powers of x cancel out: (1+x)²⁰¹ + (1-x)²⁰¹ = 2 ( ²⁰¹C₀ + ²⁰¹C₂x^2 + ²⁰¹C₄x^4