Let a1,a2,... be a strictly increasing sequence of positive real numbers such that a1=1,a2=4, and that for every positive integer k, the subsequence a4k−3,a4k−2,a4k−1,a4k is geometric and the subsequence a4k−1,a4k,a4k+1,a4k+2 is arithmetic. For each positive integer k, let rk be the common ratio of the geometric sequence a4k−3,a4k−2,a4k−1,a4k. Compute
k=1∑∞(rk−1)(rk+1−1)