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 Hint 2 by K Sengupta)

So by your logic

P(n) = 1+(n+2)²!/(n+1)²!
and Q(n)= - (n+2)²!/(n+1)²!

which is an entirely unsatisfying answer especailly since these are not polynomials (they are more like n^n) unless you consider n to be fixed in which case they are just numbers.  Admittedly they are integers so there is something here but it has little to do with polynomials.  I can fairly easily proove that no true polynomials could possibly solve this problem because f grows so quickly.

Posted by Joel on 2007-01-22 10:13:51