I am trying to prove a certain claim about a recursive sequence. The sequence is defined as follows:
$$ S_{n}^{k}(a) = \begin{cases} \sqrt{a} & \text{if } n = 1, \quad a \in \mathbb{Z}^{+}, \, a \neq 1 \\ \sum_{i=1}^{k} \sqrt{d_i} & \text{if } n > 1, \quad k \in \mathbb{Z}^{+} \end{cases} $$
where $d_i$ represents the $i$-th digit of the previous term $S_{n-1}^{k}(a)$, and $k$ represents the number of digits of $S_{n-1}^{k}(a)$ from which we extract and sum the square roots.
The claim: $\forall a$, when $k \geq 17$, $S_{n}^{k}(a) = S_{n}^{k+m}(a)$, where $m \in \mathbb{Z}^{+}$.
My question would be: How would I go on to prove or disprove this claim? Any suggestions are greatly appreciated!
Thank you!
(I am not a native English speaker, so I might be using the incorrect terms).