\(\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{x\left(x+1\right)}=\frac{996}{997}\)

\(\Rightarrow1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{996}{997}\)

\(\Rightarrow1-\frac{1}{x+1}=\frac{996}{997}\)

\(\Rightarrow\frac{1}{x+1}=\frac{1}{997}\)

\(\Rightarrow x+1=997\)

\(\Rightarrow x=996\)

\(\Leftrightarrow\)1-1/2+1/2-1/3+1/3-1/4+..+1/x-1/(x+1)=996/997

\(\Leftrightarrow\)1-1/(x+1)=996/997

\(\Leftrightarrow\)\(\frac{x}{x+1}\)\(=\frac{996}{997}\)

\(\Leftrightarrow x=996\)

\(\frac{1}{1x2}+\frac{1}{2x3}+\frac{1}{3x4}+...+\frac{1}{Xx\left(X+1\right)}=\frac{996}{997}\)

Xét dạng tổng quát : \(\frac{1}{a}-\frac{1}{a+1}=\frac{a+1}{ax\left(a+1\right)}-\frac{a}{ax\left(a+1\right)}=\frac{a+1-a}{ax\left(a+1\right)}=\frac{1}{ax\left(a+1\right)}\)

Do đó \(\frac{1}{ax\left(a+1\right)}=\frac{1}{a}-\frac{1}{a+1}\)

Áp dụng \(\frac{1}{1x2}=\frac{1}{1x\left(1+1\right)}=\frac{1}{1}-\frac{1}{1+1}=\frac{1}{1}-\frac{1}{2}\)

\(\frac{1}{2x3}=\frac{1}{2x\left(2+1\right)}=\frac{1}{2}-\frac{1}{2+1}=\frac{1}{2}-\frac{1}{3}\)

.......

\(\frac{1}{Xx\left(X+1\right)}=\frac{1}{X}-\frac{1}{X+1}\)

Do đó \(\frac{1}{1x2}+\frac{1}{2x3}+\frac{1}{3x4}+...+\frac{1}{Xx\left(X+1\right)}=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{X}-\frac{1}{X+1}\)

\(=\frac{1}{1}-\frac{1}{X+1}=1-\frac{1}{X+1}=\frac{X+1}{X+1}-\frac{1}{X+1}=\frac{X}{X+1}\)

Vì \(\frac{1}{1x2}+\frac{1}{2x3}+\frac{1}{3x4}+...+\frac{1}{Xx\left(X+1\right)}=\frac{996}{997}\)

nên \(\frac{X}{X+1}=\frac{996}{997}\)

\(\frac{X}{X+1}=\frac{996}{996+1}\)

Vậy X=996

\(\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+...+\frac{1}{y\times\left(y+1\right)}=\frac{996}{997}\)

\(\Leftrightarrow1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+....+\frac{1}{y}-\frac{1}{y+1}=\frac{996}{997}\)

\(\Leftrightarrow1-\frac{1}{y+1}=\frac{996}{997}\)

\(\Leftrightarrow\frac{1}{y+1}=1-\frac{996}{997}=\frac{1}{997}\)

\(\Leftrightarrow y+1=997\Leftrightarrow y=996\)

Vậy y = 996

1/1×2 + 1/2×3 + 1/3×4 + ... + 1/ y x (y+1) =996/997

1-1/2+1/2-1/3+1/3-1/4+...+1/y - 1/y+1 =996/997

1-1/y+1=996/997

1/ y+1 =1-996/997

1/y+1 = 997/997-996/997

1/y+1=1/997

=> y+1 =997

y=997-1

y=996

Vậy y = 996

1/1.2 + 1/2.3 + ... + 1/x.(x+1) = 996/997

1 - 1/2 + 1/2 - 1/3 + ... + 1/x - 1/x+1 = 996/997

1 - 1/x+1 = 996/997

1/x+1 = 1 - 996/997

1/x+1 = 1/997

=> x + 1 = 997

          x = 997 - 1

          x = 996

Vậy x = 996

\(=>1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{996}{997}\)

\(=>1-\frac{1}{x+1}=\frac{996}{997}\)

\(=>\frac{x+1-1}{x+1}=\frac{996}{997}\)

\(=>\frac{x}{x+1}=\frac{996}{996+1}\)

=>x=996

\(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{x\left(x+1\right)}=\frac{996}{997}\)

\(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{996}{997}\)

\(1-\frac{1}{x+1}=\frac{996}{997}\)

         \(\frac{1}{x+1}=1-\frac{996}{997}\)

          \(\frac{1}{x+1}=\frac{1}{997}\)

=> x + 1 = 997

=>       x = 997 - 1

=>       x = 996

Được chưa bây giờ tic đúng đi

\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{x.\left(x+1\right)}=\frac{996}{997}\)

\(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{996}{997}\)

\(1-\frac{1}{x+1}=\frac{996}{997}\)

         \(\frac{1}{x+1}=1-\frac{996}{997}\)

         \(\frac{1}{x+1}=\frac{1}{997}\)

\(\Rightarrow x+1=997\)

               \(x=997-1\)

               \(x=996\)

Ấm đầu hả phải tic bài làm chứ 

b) \(\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+...+\frac{1}{2013.2015}\)

\(=\frac{1}{2}\left(\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+...+\frac{2}{2013.2015}\right)\)

\(=\frac{1}{2}\left(\frac{3-1}{1.3}+\frac{5-3}{3.5}+\frac{7-5}{5.7}+...+\frac{2015-2013}{2013.2015}\right)\)

\(=\frac{1}{2}\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{2013}-\frac{1}{2015}\right)\)

\(=\frac{1}{2}\left(1-\frac{1}{2015}\right)=\frac{1007}{2015}\)

Phương trình tương đương với: 

\(\frac{1007X}{2015}=\frac{4}{2015}\Leftrightarrow X=\frac{4}{1007}\)

c) \(\frac{x+1}{2015}+\frac{x+2}{2016}=\frac{x+3}{2017}+\frac{x+4}{2018}\)

\(\Leftrightarrow\frac{x+1}{2015}-1+\frac{x+2}{2016}-1=\frac{x+3}{2017}-1+\frac{x+4}{2018}-1\)

\(\Leftrightarrow\frac{x-2014}{2015}+\frac{x-2014}{2016}=\frac{x-2014}{2017}+\frac{x-2014}{2018}\)

\(\Leftrightarrow x-2014=0\)

\(\Leftrightarrow x=2014\)

Bài này đề bài là giải phương trình hở bạn :

Gỉai 

Phương trình đẫ cho trên đề bài tương đương với :

\(\frac{x+1}{1000}+1+\frac{x+2}{999}+1+\frac{x+3}{998}+1+\frac{x+4}{997}+1+\frac{x+5}{996}+1+\frac{x+6}{995}+1=0\)

\(\Leftrightarrow\frac{x+1001}{1000}+\frac{x+1001}{999}+\frac{x+1001}{998}+\frac{x+1001}{997}+\frac{x+1001}{996}+\frac{x+1001}{995}=0\)

\(\Leftrightarrow\left(x+1001\right)\left(\frac{1}{1000}+\frac{1}{999}+\frac{1}{998}+\frac{1}{997}+\frac{1}{996}+\frac{1}{995}\right)=0\)

\(\Leftrightarrow x=-1001\)

Vậy nghiêm của phương trình là : \(x=-1001\)

Chúc bạn học tốt !!!

hoang viet nhat

Làm thiếu giải thích 

\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+.....+\frac{1}{x\left(x+1\right)}=\frac{99}{100}\)

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

\(1-\frac{1}{x+1}=\frac{99}{100}\)

=> \(\frac{1}{x+1}=1-\frac{99}{100}=\frac{1}{100}\)

=> x+1 = 100

=> x = 100 - 1 

=> x = 99

mơ đi Nguyễn Đình Dũng