Evaluate the sum of the series _ r=1 ⁵⁰ (-1)^r r ¹⁰⁰C_ r .

Step-by-step solution

Let the given sum be S = _ r=1 ⁵⁰ (-1)^r r ¹⁰⁰C_ r . Using the identity r ^ n C_ r = n ^ n-1 C_ r-1 , we can write: r ¹⁰⁰C_ r = 100 ⁹⁹C_ r-1 Substituting this into the sum, we get: S = _ r=1 ⁵⁰ (-1)^r (100 ⁹⁹C_ r-1 ) = 100 _ r=1 ⁵⁰ (-1)^r ⁹⁹C_ r-1 Let k = r-1 . As r goes from 1 to 50 , k goes from 0 to 49 . S = 100 _ k=0 ⁴⁹ (-1)^ k+1 ⁹⁹C_ k = -100 _ k=0 ⁴⁹ (-1)^k ⁹⁹C_ k We know the identity for the partial alternating sum of binomial coefficients: _ k=0 ^ m (-1)^k ^ n C_ k = (-1)^m ^ n-1 C_ m Here, n = 99 and m = 4