The coefficient of x²⁰ in the expansion of _ k=0 ⁶⁰ (1+x)^ 60-k (1-x)^k is:

Step-by-step solution

The given expression is a geometric progression with the first term a = (1+x)⁶⁰ and the common ratio r = 1-x 1+x . The number of terms in the sum is 61 . Using the sum formula for a geometric progression, S = a 1 - r^n 1 - r , we get: S = (1+x)⁶⁰ 1 - ( 1-x 1+x )⁶¹ 1 - 1-x 1+x S = (1+x)⁶⁰ (1+x)⁶¹ - (1-x)⁶¹ (1+x)⁶¹ (1+x) - (1-x) 1+x S = (1+x)⁶¹ - (1-x)⁶¹ 2x We need to find the coefficient of x²⁰ in S , which is equivalent to finding the coefficient of x²¹ in the numerator (1+x)⁶¹ - (1-x)⁶¹ 2 . The coefficient of x²¹