Greatest Integer | PRMO 2019 | Question 22

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Try this beautiful problem from the Pre-RMO, 2019 based on Greatest Integer.

Greatest integer – PRMO 2019

Find the greatest integer not exceeding the sum \(\sum_{n=1}^{1599}\frac{1}{\sqrt{n}}\)

Largest integer

Divisibility

Integer

But try the problem first…

Answer: is 78.

Source

First hint

\(\int\limits_1^{1600}\frac{1}{x}{d}x < \sum_{x=1}^{1599}\frac{1}{\sqrt{n}}\)

\(< 1+\sum_{n=1}^{1599}\frac{1}{\sqrt{x}}{d}x\)

Second Hint

or, \([2\sqrt{x}]_{1}^{1600}< \sum_{n=1}^{1599}\frac{1}{\sqrt{n}}\)

\(< 1+|2{\sqrt{x}}|_1^{1599}\)

Final Step

or, 78<\(\sum_{n=1}^{1599}\frac{1}{\sqrt{n}} <2\sqrt{1599}-1\)

or, 78 < \(\sum_{n=1}^{1599}\frac{1}{\sqrt{n}}\)<78.97

or,\([\sum_{n=1}^{1599}\frac{1}{\sqrt{n}}]\)=78.