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May 11, 2020

Greatest Integer | PRMO 2019 | Question 22

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 \(\sum_{n=1}^{1599}\frac{1}{\sqrt{n}}\)

  • is 107
  • is 78
  • is 840
  • cannot be determined from the given information

Key Concepts


Largest integer

Divisibility

Integer

Check the Answer


Answer: is 78.

PRMO, 2019, Question 22

Elementary Number Theory by David Burton

Try with Hints


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.

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