I am currently running a web scraping program and seeing the time number constantly showing up on my screen. Then suddenly I have a question as the title says.
To specify,
Given $a_1, a_2, a_3, \dots$ an arithmetic series and $b_i = \lfloor a_i\rfloor$ ($b_i$ is the biggest integer that is not larger than $a_i$). Is it possible to get the common difference $d = a_{i+1} - a_i$ ONLY from series $\{b_i\}$?
$\newcommand\fr{\operatorname{frac}}$ Here’s the proof: As $a_k = a_0 + kd$ and setting $\fr x = x - \lfloor x \rfloor$ (which always is in $[0..1)$), $$\frac {\lfloor a_k \rfloor} k = \frac {a_k - \fr a_k} k = d + \frac{a_0 + \fr a_k} k \overset{k → ∞}\longrightarrow d.$$