Adding 1 to each entry of continued fractions Here we denote $[a_0,...,a_n]$ as the continued fraction of some rational number. If I take $p/q=[a_0,a_1,...,a_n]$ to $p'/q'=[a_0+1,a_1+1,...,a_n+1]$, are there any nice properties I can say about $p'/q'$?

We have $p' = p + q$ and $q' = q$. This is because, in a continued fraction, the $n$-th term is $a_n = \lfloor \frac{p}{q} \rfloor$, the greatest integer less than or equal to $\frac{p}{q}$. But we have
$$
\frac{p}{q} = a_0 \frac{1}{1} + \frac{1}{a_1 \frac{1}{1} + \frac{1}{\ddots + \frac{1}{a_n}}}
$$
so
$$
\frac{p + 1}{q + 1} = a_0 \frac{1}{1} + \frac{1}{a_1 \frac{1}{1} + \frac{1}{\ddots + \frac{1}{a_n + \frac{1}{1}}}} = \frac{p'}{q'}
$$
where $p' = p + q$ and $q' = q$.
