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Evaluate \(\lceil\sqrt{5}\rceil + \lceil\sqrt{6}\rceil + \lceil\sqrt{7}\rceil + \cdots + \lceil\sqrt{29}\rceil\). Note: For a real number x,\(\lceil x \rceil\)  denotes the smallest integer that is greater than or equal to x. 

 Jul 29, 2020
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it equals 3+3+3+3+3 for the ones below 9 and 4+4+4+4+4+4+4 for the ones from 10 ~16 and 5+5+5+5+5+5+5+5+5 for the ones 17~25 and 6+6+6 for 26~29 when we add them together we have 3+3+3+3+3+4+4+4+4+4+4+4+5+5+5+5+5+5+5+5+5+6+6+6=106 so our final answer is \(\lceil\sqrt{5}\rceil + \lceil\sqrt{6}\rceil + \lceil\sqrt{7}\rceil + \cdots + \lceil\sqrt{29}\rceil=\boxed{106}\)

 Jul 29, 2020

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