MathDB

Multiples of 7 in the sequence 1, 12, 123, ...

Source: Brazilian Mathematical Olympiad 2018 - Q5

November 16, 2018

number theorySequencemultipleBrazilian Math OlympiadBrazilian Math Olympiad 2018

Problem Statement

Consider the sequence in which

a_1 = 1

and

a_n

is obtained by juxtaposing the decimal representation of

n

at the end of the decimal representation of

a_{n-1}

. That is,

a_1 = 1

a_2 = 12

a_3 = 123

\dots

a_9 = 123456789

a_{10} = 12345678910

and so on. Prove that infinitely many numbers of this sequence are multiples of

7