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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

inequality

Integers

Algebra

## Check the Answer

But try the problem first…

Source

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

$\Rightarrow 29 values.$