This is a question about prime gaps $g_n = p_{n+1}-p_n$ that started with a look at the average of ratios $$r_n=\frac{p_{n+1}-p_n}{p_{2n+1}-p_{2n}}$$ and of the inverse, $$ s_n=\frac{p_{2n+1}-p_{2n}}{p_{n+1}-p_{n}},$$ $n$ running from 1 to k. The averages are $$a_r=\frac{1}{k}\sum_{n=1}^k r_n,~~a_s=~\frac{1}{k}\sum_{n=1}^k s_n.$$
My questions were: (1) Is $a_r \sim a_s$? (2) Does $a_r$ ever exceed $a_s?$
What I did (can skip): For (1), FWIW, it seems that both averages may be unbounded as $k$ grows large. I think that the PNT gives (1) via $2p_k\sim p_{2k}$ and a bit more work. For (2), we have the same number of primes on $[p_1, p_{k}]$ as on $[p_{k+1},p_{2k}].$ The slow but jumpy growth of gaps gives $a_s$ an advantage at first, but as $k$ gets large I think the absolute difference shrinks and there may be (many) $k$ for which $a_s<a_r.$ However such a $k$ is greater than 100,000, for which $a_r\approx 1.866$ and $a_s\approx 2.02.$