Question

a) So sánh: \(\sqrt{17}\)+ \(\sqrt{26}\)+1   và   \(\sqrt{99}\)

b) CMR: \(\frac{1}{\sqrt{1}}\)+\(\frac{1}{\sqrt{2}}\)+\(\frac{1}{\sqrt{3}}\)+....+\(\frac{1}{\sqrt{100}}\)>10

Answer 1

a) Ta có \(\sqrt{17}\)>\(\sqrt{16}\)

             \(\sqrt{26}\)>\(\sqrt{25}\)

=>\(\sqrt{17}\)+\(\sqrt{26}\)+1>\(\sqrt{16}\)+\(\sqrt{25}\)+1

=>\(\sqrt{17}\)+\(\sqrt{26}\)+1> 4+ 5 +1

=>\(\sqrt{17}\)+\(\sqrt{26}\)+1 >10 hay >\(\sqrt{100}\)

=>\(\sqrt{17}\)+\(\sqrt{26}\)+1>\(\sqrt{99}\)

b) \(\frac{1}{\sqrt{1}}\)=1 >\(\frac{1}{10}\)

    \(\frac{1}{\sqrt{2}}\)>\(\frac{1}{\sqrt{100}}\)=\(\frac{1}{10}\)

....................................

   \(\frac{1}{\sqrt{100}}\)=\(\frac{1}{10}\)

=>\(\frac{1}{\sqrt{1}}\)+\(\frac{1}{\sqrt{2}}\)+\(\frac{1}{\sqrt{3}}\)+...+\(\frac{1}{\sqrt{100}}\)>\(\frac{1}{10}\)+\(\frac{1}{10}\)+...+\(\frac{1}{10}\)(có 100 số \(\frac{1}{10}\))

=>\(\frac{1}{\sqrt{1}}\)+\(\frac{1}{\sqrt{2}}\)+\(\frac{1}{\sqrt{3}}\)+...+\(\frac{1}{\sqrt{100}}\)> \(\frac{100}{10}\)=10 

Answer 2

\(a)\) Ta có : 

\(\sqrt{17}+\sqrt{26}+1>\sqrt{16}+\sqrt{25}+1=4+5+1=10=\sqrt{100}>\sqrt{99}\)

Vậy \(\sqrt{17}+\sqrt{26}+1>\sqrt{99}\)

Chúc bạn học tốt ~