Try this beautiful problem from the Pre-RMO, 2017 based on Integers and Inequality.

Integers and Inequality – PRMO 2017


Find the number of positive integers n such that \(\sqrt{n}+\sqrt{n+1} \lt 11\)

  • is 107
  • is 29
  • is 840
  • cannot be determined from the given information

Key Concepts


inequality

Integers

Algebra

Check the Answer


But try the problem first…

Answer: is 29.

Source
Suggested Reading

PRMO, 2017, Question 7

Elementary Algebra by Hall and Knight

Try with Hints


First hint

here \(\sqrt{n}+\sqrt{n+1} \lt 11\) for n=1,2,3,4,5,6,7,8,….,16,…..25

taking \(\sqrt{n}+\sqrt{n+1}=11\) is first equation

Second Hint

\(\Rightarrow \frac{1}{\sqrt{n}+\sqrt{n+1}}=\frac{1}{11}\)

\(\Rightarrow \sqrt{n+1}-\sqrt{n}=\frac{1}{11}\) is second equation

adding both equations \(2\sqrt{n+1}\)=\(\frac{122}{11}\)

Final Step

\(\Rightarrow n+1 = \frac{3721}{121}\)

\(\Rightarrow n=\frac{3600}{121}\)

=29.75

\(\Rightarrow 29 values.\)

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