Competition problem — Number theory

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Number theory Difficulty 5.6 AIME, harder Prove it

## Subiectul I.(20 puncte )

Se consideră şirul de numere $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, \ldots$ , unde $a_{1}=1, a_{2}=3, a_{3}=7, a_{4}=15, a_{5}=31, \ldots$ Să se arate că $a_{2014}+1$ este pătrat perfect.

prof. Măgdaş Elena, Şcoala Gimnazială "Horea” Cluj-Napoca

Solution

## Subiectul I.

$a_{1}=2^{1}-1, a_{2}=2^{2}-1, a_{3}=2^{3}-1, a_{4}=2^{4}-1, \ldots a_{2014}=2^{2014}-1 \Rightarrow a_{2014}+1=2^{2014}=\left(2^{1007}\right)^{2}$

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.