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# Integers and Inequality | PRMO 2017 | Question 7

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

PRMO, 2017, Question 7

Elementary Algebra by Hall and Knight

## Try with Hints

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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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

PRMO, 2017, Question 7

Elementary Algebra by Hall and Knight

## Try with Hints

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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