Let
,
,
,
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,
. Prove that
for all
.
Proposed by Morteza Saghafian, Iran
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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, $\gcd (a_i, a_{i + 1}) > a_{i - 1}$. Prove that $a_n\ge 2^n$ for all $n\ge 0$.
Proposed by Morteza Saghafian, Iran