CM cai j cay

//Chứng minh gì thế=)) < 7/12 hay > 7/12 hay = 7/12

A=1-1/2+1/3-1/4+.....+1/99-1/100

   =(1+1/3+1/5+...+1/99)-(1/2+1/4+...+1/100)

    =(1+1/3+...+1/99)-2(1/2+1/4+...+1/100)

    =(1+1/3+...+1/99)-(1+1/2+...+1/50)

...............TỰ TÍNH TIẾP HEM

=>A=7/12 ĐÚNG KO

\(\frac{1}{1.2}+\frac{1}{3.4}+\frac{1}{5.6}+...+\frac{1}{99.100}\)

\(=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{99}-\frac{1}{100}\)

\(=\left(1+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{99}\right)-\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{100}\right)\)

\(=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...+\frac{1}{99}+\frac{1}{100}\right)-2.\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{100}\right)\)

\(=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...+\frac{1}{99}+\frac{1}{100}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{50}\right)\)

\(=\frac{1}{51}+\frac{1}{52}+...+\frac{1}{100}\)

Ta có:

\(\frac{1}{51}>\frac{1}{75}\)

\(\frac{1}{52}>\frac{1}{75}\)

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

\(\frac{1}{75}=\frac{1}{75}\)

\(\Rightarrow\frac{1}{51}+\frac{1}{52}+...+\frac{1}{75}>\frac{1}{75}+\frac{1}{75}+...+\frac{1}{75}=25.\frac{1}{75}=\frac{1}{3}\)(1)

Ta có:

\(\frac{1}{76}>\frac{1}{100}\)

\(\frac{1}{77}>\frac{1}{100}\)

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

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

\(\Rightarrow\frac{1}{76}+\frac{1}{77}+...+\frac{1}{100}>\frac{1}{100}+\frac{1}{100}+...+\frac{1}{100}=25.\frac{1}{100}=\frac{1}{4}\)(2)

Từ (1) và (2) ta có:

\(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{75}+\frac{1}{76}+...+\frac{1}{100}>\frac{1}{3}+\frac{1}{4}=\frac{7}{12}\)

\(\Rightarrow\frac{1}{51}+\frac{1}{52}+...+\frac{1}{100}>\frac{7}{12}\)(5)

Ta có:

\(\frac{1}{51}=\frac{1}{51}\)

\(\frac{1}{52}< \frac{1}{51}\)

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

\(\frac{1}{75}< \frac{1}{51}\)

\(\Rightarrow\frac{1}{51}+\frac{1}{52}+...+\frac{1}{75}< \frac{1}{51}+\frac{1}{51}+...+\frac{1}{51}=25.\frac{1}{51}< 25.\frac{1}{50}=\frac{1}{2}\)(3)

Ta có:

\(\frac{1}{76}=\frac{1}{76}\)

\(\frac{1}{77}< \frac{1}{76}\)

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

\(\frac{1}{100}< \frac{1}{76}\)

\(\Rightarrow\frac{1}{76}+\frac{1}{77}+...+\frac{1}{100}< \frac{1}{76}+\frac{1}{76}+...+\frac{1}{76}=25.\frac{1}{76}< 25.\frac{1}{75}=\frac{1}{3}\)(4)

Từ (3) và (4) ta có:

\(\frac{1}{51}+\frac{1}{52}+...+\frac{1}{75}+\frac{1}{76}+...+\frac{1}{100}>\frac{1}{2}+\frac{1}{3}=\frac{5}{6}\)

\(\Rightarrow\frac{1}{51}+\frac{1}{52}+...+\frac{1}{100}< \frac{5}{6}\)(6)

Từ (5) và (6) 

\(\Rightarrow\frac{7}{12}< \frac{1}{1.2}+\frac{1}{3.4}+...+\frac{1}{99.100}< \frac{5}{6}\)

                                                            đpcm

Tham khảo nhé~

ta có: A=\(\frac{1}{51}+\frac{1}{52}+\frac{1}{53}+...+\frac{1}{100}=>A=\left(\frac{1}{51}+\frac{1}{52}+\frac{1}{53}+..+\frac{1}{75}\right)+\left(\frac{1}{76}+\frac{1}{77}+...+\frac{1}{100}\right)\)

\(\frac{1}{51}>\frac{1}{52}>\frac{1}{53}>...>\frac{1}{75};\frac{1}{76}>\frac{1}{77}>...>\frac{1}{100}\)

do đó:\(A>\frac{1}{75}.25+\frac{1}{100}.25=>A>\frac{1}{3}+\frac{1}{4}=\frac{7}{12}\)  (1)

lại có: \(A<\frac{1}{51}.25+\frac{1}{76}.25<\frac{1}{50}.25+\frac{1}{75}.25=\frac{1}{2}+\frac{1}{3}=\frac{5}{6}\)   (2)

từ (1) và (2)=>7/12<A<5/6(đpcm)

Ta có

\(A=\frac{2-1}{1.2}+\frac{4-3}{3.4}+\frac{6-5}{5.6}+...+\frac{100-99}{99.100}\)

\(A=\frac{2}{1.2}-\frac{1}{1.2}+\frac{4}{3.4}-\frac{3}{3.4}+\frac{6}{5.6}-\frac{5}{5.6}+...+\frac{100}{99.100}-\frac{99}{100.99}\)

\(A=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{99}-\frac{1}{100}\)

<=>A=1-1/100=99/100

=>7/12<A<5/6(bấm máy tính là biết)

Số chia rút gọn thành 1/51+1/52+...+1/99+1/100

=> biểu thức bằng 1