Question

Using induction or otherwise, prove that for any non-negative integers m, n, r and k

∑(n - m) (r + m)!/m!, m ∈ [m = 0, k] = (r + k + 1)!/k![n/r + 1 - k/r + 2]

  

Answer 1

Let P(k):

For k = 1, we will have two terms on LHS, in summation for m = 0 and m = 1, so that

Hence, the result is true for k = s + 1 and so by mathematical induction, the formula holds for all k ∈ N.