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$$S_k=\sum_{n=1}^{k^2-1}\lfloor\sqrt{n}\rfloor $$

Can somebody give me an idea about the steps I should follow? Initially I thought
$$n^{1/2}\log(n) \leq n^{1/2}\leq n^{3/2}$$

so $\Theta(f(n))=S_k $ but I am not sure if it is the strict bound.

Thanks

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