Friday, July 10, 2015

Consecutive product sum [Solution]

The task was to find (without pen and paper)

nk=rk(k1)(kr)


Solution


Let Sk=k(k1)(kr).

One way to solve this is to notice that

(k+1)k(k1)(kr)k(k1)(k2)(kr)(kr+1)


=(k+1(kr+1))k(k1)(kr)=rk(k1)(kr)


i.e

PkPk1=rSk


where Pk=(k+1)k(k1)(kr)

Thus we get a sum where most of the terms cancel each other out (called a telescopic sum) and the answer is simply PnPr1 which is

(n+1)n(n1)(nr)r


There are other solutions, like writing in terms of binomial coefficients etc, not sure we can do those without pen and paper though.

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