Central Binomials Appear

An arithmetic sequence $\{a_n\}$ with nonzero common difference satisfies $a_1 = 2$, and $a_1, a_3, a_7$ form a geometric sequence.

1. Find a closed formula for $a_n$. 2. A sequence $\{b_n\}$ satisfies $b_1 = 1$ and $b_nb_{n+1} = a_n$.

1. Prove that $b_{2n} = \dfrac{4^n}{\binom{2n}{n}}$.
2. Does there exist $n \in \mathbb{N}^*$ such that

Justify your answer.