Question

Tìm số nguyên dương n nhỏ nhất thỏa mãn: \(\frac{1}{\sqrt{n}+\sqrt{n+1}}+\frac{1}{\sqrt{n+1}+\sqrt{n+2}}< 0,02\)

Answer 1

\(\Leftrightarrow\sqrt{n+1}-\sqrt{n}+\sqrt{n+2}-\sqrt{n+1}< \frac{1}{50}\)

\(\Leftrightarrow\sqrt{n+2}-\sqrt{n}< \frac{1}{50}\)

\(\Leftrightarrow\frac{2}{\sqrt{n+2}+\sqrt{n}}< \frac{1}{50}\)

\(\Leftrightarrow\sqrt{n+2}+\sqrt{n}>100\)

\(\Leftrightarrow\sqrt{n+2}>100-\sqrt{n}\)

Với \(n\le10000\) \(\Rightarrow n+2>100^2+n-200\sqrt{n}\)

\(\Rightarrow200\sqrt{n}>9998\)

\(\Rightarrow\sqrt{n}>\frac{4999}{100}\Rightarrow n>49,99^2\Rightarrow n_{min}=50^2=2500\)