2008 IMO Shortlist N3

Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $\operatorname{gcd}\left(a_{i}, a_{i+1}\right)>a_{i-1}$. Prove that $a_{n} \geq 2^{n}$ for all $n \geq 0$.

令 $a_{0}, a_{1}, a_{2}, \ldots$ 为正整数序列，使得任意两个连续项的最大公约数大于前一项；在符号中，$\operatorname{gcd}\left(a_{i}, a_{i+1}\right)>a_{i-1}$。证明 $a_{n} \geq 2^{n}$ 对于所有 $n \geq 0$。