Seductive Recursive Sequences

Algebra Level 2

(αn)(\alpha_n) and (βn)(\beta_n) are two sequences defined recursively as αn=2αn1+1\alpha_n=2 \alpha_{n-1}+1 and βn=8βn1+1\beta_n=8 \beta_{n-1}+1 with α0=β0=0.\alpha_0=\beta_0=0. Find the ratio

α2016β672. \frac{\alpha_{2016}}{\beta_{672}}.

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