IMO 1977 LL FIN43

Origin: FIN

Problem

Evaluate S = n  k=1 k(k + 1) \cdot \cdot \cdot (k + p), where n and p are positive integers.

Solution

Since k(k + 1) \cdot \cdot \cdot (k + p) = (p + 1)!
k+p p+1  = (p + 1)! 
k+p+1 p+2  −
k+p p+2  , it follows that n  k=1 k(k + 1) \cdot \cdot \cdot (k + p) = (p + 1)! n + p + 1 p + 2  = n(n + 1) \cdot \cdot \cdot (n + p + 1) p + 2 .