Let n be a positive integer and let f(n)= 1²!+ 2²!+ 3²!+...+n²!

Determine polynomials P(n) and Q(n) such that f(n+2)= P(n)f(n+1)+ Q(n)f(n).

(In reply to The Facts are out by Gamer)

f(n+1) - f(n)
=(n+1)²!
= (n^2 + 2n + 1)!
f(n+2) - f(n+1)
=(n+2)²!
= (n^2 + 4n +4)!
= (n^2 + 2n + 2)(n^2 + 2n + 3)..........(n^2 + 4n+4) (f(n+1) - f(n))
Accordigly, it would now be feasible to determine polynomials
P(n) and Q(n) satisfying conditions of the given problem.

Edited on January 22, 2007, 4:27 am

Posted by K Sengupta on 2007-01-22 04:26:43