MathDB

sequence, prove periodicity and existence of sufficiently small element

Source: French MO 1996 P2

April 11, 2021

algebraSequence

Problem Statement

Let

a

be an odd natural number and

b

be a positive integer. We define a sequence of reals

(u_n)

as follows:

u_0=b

and, for all

n\in\mathbb N_0

u_{n+1}

\frac{u_n}2

u_n

is even and

a+u_n

otherwise.
(a) Prove that one can find an element of

u_n

smaller than

a

. (b) Prove that the sequence is eventually periodic.