Question

cmr \(\frac{1}{\sqrt{1}}\)+ \(\frac{1}{\sqrt{2}}\)+\(\frac{1}{\sqrt{3}}\)+........+\(\frac{1}{\sqrt{25}}\)> 5

 

Answer 1

\(\frac{1}{\sqrt{1}}=1\)

\(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}>\frac{1}{\sqrt{4}}+\frac{1}{\sqrt{4}}=\frac{2}{\sqrt{4}}=1\)

\(\frac{1}{\sqrt{4}}+\frac{1}{\sqrt{5}}+\frac{1}{\sqrt{6}}+\frac{1}{\sqrt{7}}+\frac{1}{\sqrt{8}}>5\cdot\frac{1}{\sqrt{9}}=\frac{5}{3}>1\)

\(\frac{1}{\sqrt{9}}+\frac{1}{\sqrt{10}}+\frac{1}{\sqrt{11}}+...+\frac{1}{\sqrt{15}}>7\cdot\frac{1}{\sqrt{16}}=\frac{7}{4}>1\)

\(\frac{1}{\sqrt{16}}+\frac{1}{\sqrt{17}}+...+\frac{1}{\sqrt{25}}>10\cdot\frac{1}{\sqrt{25}}=2>1\)

Cộng từng vế ta có:

\(\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{25}}>5\)đpcm