$\gcd(a_n ,a_{n+k})< \frac{a_k}2$ for a sequence such that $a_{k+2}=a_{k+1}+a_k$ and $\gcd(a_1,a_2)=1$

Question

How to solve:

Consider a sequence $(a_{k})_{k\geq 1}$ of natural number defined as follow: $a_1=a$ and $a_2=b$ with $a,b> 1$ and $\gcd(a,b)=1$ for all $k$ >$0$, $a_{k+2}=a_{k+1}+a_k$. Prove that for all natural numbers $n$ and $k$, $\gcd(a_n ,a_{n+k})< \dfrac{a_k}{2}$.

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