I got this question in an exam (high school level) I recently appeared in.
If $ \mathop{\sum}\limits_{{k}{=}{1}}\limits^{80}{\frac{1}{\sqrt{k}}}{=}{S} $, then find the greatest integer less than or equal to $S$.
This is supposed to be a problem from 'sequences and series' so I tried converting the sum into a telescopic sum, but to no avail. The closest I've come to solving it is by taking a completely different approach. I evaluated $ \mathop{\int}\limits_{1}\limits^{80}{\frac{1}{\sqrt{x}}}dx $ to put a lower bound to the sum. This value comes out to be approximately 15.8, but the answer to the problem is 16 so this isn't really helpful. Anyhow, I feel like I am nowhere near solving this question so any help would be greatly appreciated.
Thank you.