MathDB

exists c : b_n=c a_n. a_{n+2}=a_n+a_{n+1}, b_{n+2}=b_n+b_{n+1}, a_n / b_n

Source: 2021 Francophone MO Seniors p1

April 3, 2021

Sequencenumber theorydividesrecurrence relationFrancophone

Problem Statement

Let

a_1,a_2,a_3,\ldots

and

b_1,b_2,b_3,\ldots

be positive integers such that

a_{n+2} = a_n + a_{n+1}

and

b_{n+2} = b_n + b_{n+1}

for all

n \ge 1

. Assume that

a_n

divides

b_n

for infinitely many values of

n

. Prove that there exists an integer

c

such that

b_n = c a_n

for all

n \ge 1