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Two linked sequences

Source: 2022 Nigerian Senior MO Round 2/Problem 4

May 6, 2022

algebraSequenceTwo sequences

Problem Statement

Define sequence

(a_{n})_{n=1}^{\infty}

by

a_1=a_2=a_3=1

and

a_{n+3}=a_{n+1}+a_{n}

for all

n \geq 1

. Also, define sequence

(b_{n})_{n=1}^{\infty}

by

b_1=b_2=b_3=b_4=b_5=1

and

b_{n+5}=b_{n+4}+b_{n}

for all

n \geq 1

. Prove that

\exists N \in \mathbb{N}

such that

a_n = b_{n+1} + b_{n-8}

for all

n \geq N

.

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