On geometric progressions on hyperelliptic curves

Let $C$ be a hyperelliptic curve over $\mathbb Q$ described by $y^2=a_0x^n+a_1x^{n-1}+\ldots+a_n$, $a_i\in\mathbb Q$. The points $P_{i}=(x_{i},y_{i})\in C(\mathbb{Q})$, $i=1,2,...,k,$ are said to be in a geometric progression of length $k$ if the rational numbers $x_{i}$, $i=1,2,...,k,$ form a geometric progression sequence in $\mathbb Q$, i.e., $x_i=pt^{i}$ for some $p,t\in\mathbb Q$. In this paper we prove the existence of an infinite family of hyperelliptic curves on which there is a sequence of rational points in a geometric progression of length at least eight.