Integers and Inequality | PRMO 2017 | Question 7

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

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

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

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

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

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

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

=29.75

\(\Rightarrow 29 values.\)

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