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u_{n+1} =u_n(u_n + 1)/ n, rational numbers

Source: Austrian Regional Competition For Advanced Students 2000 p4

October 4, 2022

rationalnumber theoryalgebraSequencerecurrence relation

Problem Statement

We consider the sequence

\{u_n\}

defined by recursion

u_{n+1} =\frac{u_n(u_n + 1)}{n}

for

n \ge 1

. (a) Determine the terms of the sequence for

u_1 = 1

. (b) Show that if a member of the sequence is rational, then all subsequent members are also rational numbers. (c) Show that for every natural number

K

there is a

u_1 > 1

such that the first

K

terms of the sequence are natural numbers.

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