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I'm trying to evaluate and prove a simple statement but It seems really raw/bad solution. I would like to advise with you if this is the right way because It is really getting more complicated than It should be.

Suppose I have the following statement:

$ \sum\limits_{i=0}^{\sqrt{n}} \sqrt{i} = \theta(n) $

So I had a small exertion between three methods/technique to evaluate this kind of sum:

  1. Simplely plug in the ranges.

  2. Split the sum into two different sums and evaluate the desireable bound.

  3. Using the intergral statement to evaluate the bound of sums.

I believe the second option is correct but I'm getting a "wicked" expression and I'm not sure It is the correct way to solve this issue.

My final expression:

$ \sum\limits_{i=0}^{\sqrt{n}} \sqrt{i} \ge \frac{\sqrt{n}}{2} * \sqrt{1} + \frac{\sqrt{n}}{2} * \sqrt{\frac{\sqrt{n}}{2} + 1} $

Am I on the right track?

Thank you guys, Syndicator.

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