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

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

inequality

Integers

Algebra

But try the problem first...

Answer: is 29.

Source

Suggested Reading

PRMO, 2017, Question 7

Elementary Algebra by Hall and Knight

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

\(\Rightarrow 29 values.\)

- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA

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