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Olympiad workbook
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Olympiad workbook
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IOQM
Chapter
Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

Find the number of positive integers n such that n + n+1 < 11 .

Find the number of positive integers nn such that n+n+1<11\sqrt{n}+\sqrt{n+1} < 11.

Correct answer

29

Step-by-step solution

N+N+1<11 N=1,2,3,4,5,6,7,8…….16……25 N+1+N=11...(1)1 N+1−N=11 N+1−N=111...(2)2 N+1=12211⇒ N+1=6111⇒ N+1=3721121 N=3600121=29.75\begin{aligned} & \sqrt{\mathrm{N}}+\sqrt{\mathrm{N}+1} < 11 \\ & \mathrm{~N}=1,2,3,4,5,6,7,8 \ldots \ldots .16 \ldots \ldots 25 \\ & \sqrt{\mathrm{~N}+1}+\sqrt{\mathrm{N}}=11 ...(1)\\ & \frac{1}{\sqrt{\mathrm{~N}+1}-\sqrt{\mathrm{N}}}=11 \\ & \sqrt{\mathrm{~N}+1}-\sqrt{\mathrm{N}}=\frac{1}{11} ...(2)\\ & 2 \sqrt{\mathrm{~N}+1}=\frac{122}{11} \Rightarrow \sqrt{\mathrm{~N}+1}=\frac{61}{11} \Rightarrow \mathrm{~N}+1=\frac{3721}{121} \\ & \mathrm{~N}=\frac{3600}{121}=29.75 \end{aligned} So 29 values
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