If ∑ k = 0 100 k k + 1 C k 100 = a ⋅ 2 100 + b c , where a , b , c ∈ N , then the least value of a + b + c 100 is equal to

Step-by-step solution

Let S = ∑ k = 0 100 k k + 1 C 100 k = ∑ k = 0 100 k + 1 - 1 k + 1 C 100 k = ∑ k = 0 100 C 100 k - ∑ k = 0 100 1 k + 1 C 100 k = 2 100 - 1 101 ∑ k = 0 100 101 k + 1 C 100 k = 2 100 - 1 101 ∑ k = 0 100 C 101 k + 1 = 2 100 - 1 101 2 101 - 1 = 101 ⋅ 2 100 − 2 101 + 1 101 = 99 ⋅ 2 100 + 1 101 So, a = 99 ,   b = 1 ,   c = 101 Hence, a + b + c 100 least = 2 . 01