Question

Chứng tỏ rằng : \(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+..+\frac{1}{17}< 2\)

Answer 1

Đặt \(A=\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+....+\frac{1}{17}\)

Ta có: \(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}< \frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}=\frac{6}{5}\left(1\right)\)

\(\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{1}{15}+\frac{1}{16}+\frac{1}{17}< \frac{1}{11}+\frac{1}{11}+\frac{1}{11}+\frac{1}{11}+\frac{1}{11}+\frac{1}{11}+\frac{1}{11}=\frac{7}{11}\left(2\right)\)

Từ (1)(2) \(\Rightarrow A< \frac{6}{5}+\frac{7}{11}=\frac{66}{55}+\frac{35}{55}=\frac{101}{55}< \frac{110}{55}=2\)

\(\Rightarrow A< 2\Rightarrow\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+...+\frac{1}{17}< 2\left(đpcm\right)\)

Answer 2

Ta có:

1/5=1/5

1/6<1/5

1/7<1/5

..........

1/10<1/5

=>1/5+1/6+...+1/10<1/5.6=6/5(1)

Lại có :

1/11=1/11

1/12<1/11

1/13<1/11

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

1/17<1/11

=>1/11+1/12+1/13+...+1/17<1/11.7=7/11(2)

Từ (1)và (2)=>1/5+1/6+...+1/17<6/5+7/11=101/55<110/55=2

=>1/5+1/6+...+1/17<2

ĐPCM