Let P(x) = (1+x)¹⁰⁰ + x(1+x)⁹⁹ + x^2(1+x)⁹⁸ + + x¹⁰⁰ . The sum of the coefficients of x, x^2, x^3, , x⁵⁰ in the polynomial P(x) is :

Step-by-step solution

The given polynomial P(x) is a geometric series with first term a = (1+x)¹⁰⁰ , common ratio r = x 1+x , and 101 terms. Using the sum formula for a geometric progression: P(x) = (1+x)¹⁰⁰ 1 - ( x 1+x )¹⁰¹ 1 - x 1+x P(x) = (1+x)¹⁰⁰ (1+x)¹⁰¹ - x¹⁰¹ (1+x)¹⁰¹ 1 1+x P(x) = (1+x)¹⁰¹ - x¹⁰¹ We need the sum of the coefficients of x, x^2, , x⁵⁰ in P(x) . Since the highest degree term subtracted is x¹⁰¹ , it does not affect the coefficients of x^k for 1 k 50 . The coefficient of x^k in (1+x)¹⁰¹ is ¹⁰¹C_ k . The required sum is