MathDB

3

Part of 1991 Brazil National Olympiad

Problems(1)

This sequence is a polynomial? a rational function?

Source: Problem 3, Brazilian MO, 1991

3/19/2006
Given k>0k > 0, the sequence ana_n is defined by its first two members and an+2=an+1+knan a_{n+2} = a_{n+1} + \frac{k}{n}a_n a)For which kk can we write ana_n as a polynomial in nn? b) For which kk can we write an+1an=p(n)q(n)\frac{a_{n+1}}{a_n} = \frac{p(n)}{q(n)}? (p,qp,q are polynomials in R[X]\mathbb R[X]).
algebrapolynomialfunctionrational functionalgebra proposed