Try this beautiful problem from the Pre-RMO, 2017 based on Integers and Inequality.
Find the number of positive integers n such that \(\sqrt{n}+\sqrt{n+1} \lt 11\)
inequality
Integers
Algebra
But try the problem first...
Answer: is 29.
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
\(\Rightarrow 29 values.\)
Try this beautiful problem from the Pre-RMO, 2017 based on Integers and Inequality.
Find the number of positive integers n such that \(\sqrt{n}+\sqrt{n+1} \lt 11\)
inequality
Integers
Algebra
But try the problem first...
Answer: is 29.
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
\(\Rightarrow 29 values.\)
