Question 97
a) If $(a_n)_{n \ge 1}$ is a strictly increasing sequence of positive integers such that $(a_{2n-1} + a_{2n})/a_n$ is constant as $n$ runs through all positive integers, then this constant is an integer greater than or equal to $4$; and
b) Given an integer $N \ge 4$, there exists a strictly increasing sequence $(a_n)_{n \ge 1}$ of positive integers such that $(a_{2n-1} + a_{2n})/a_n = N$ for all indices $n$.
b) Given an integer $N \ge 4$, there exists a strictly increasing sequence $(a_n)_{n \ge 1}$ of positive integers such that $(a_{2n-1} + a_{2n})/a_n = N$ for all indices $n$.