MathDB

Problem 4

Part of 2022 Nigerian Senior MO Round 2

Problems(1)

Two linked sequences

Source: 2022 Nigerian Senior MO Round 2/Problem 4

5/6/2022
Define sequence (an)n=1∞(a_{n})_{n=1}^{\infty} by a1=a2=a3=1a_1=a_2=a_3=1 and an+3=an+1+ana_{n+3}=a_{n+1}+a_{n} for all n≥1n \geq 1. Also, define sequence (bn)n=1∞(b_{n})_{n=1}^{\infty} by b1=b2=b3=b4=b5=1b_1=b_2=b_3=b_4=b_5=1 and bn+5=bn+4+bnb_{n+5}=b_{n+4}+b_{n} for all n≥1n \geq 1. Prove that ∃N∈N\exists N \in \mathbb{N} such that an=bn+1+bn−8a_n = b_{n+1} + b_{n-8} for all n≥Nn \geq N.
algebraSequenceTwo sequences