Gọi tổng trên là A
A=1/1.2.3+1/2.3.4+1/3.4.5+...1/98.99.100
Ta xét :
1/1.2 ‐ 1/2.3 = 2/1.2.3; 1/2.3 ‐ 1/3.4 = 2/2.3.4;...; 1/98.99 ‐ 1/99.100 = 2/98.99.100
tổng quát: 1/n﴾n+1﴿ ‐ 1/﴾n+1﴿﴾n+2﴿ = 2/n﴾n+1﴿﴾n+2﴿.
Do đó: 2A = 2/1.2.3 + 2/2.3.4 + 2/3.4.5 +...+ 2/98.99.100
= ﴾1/1.2 ‐ 1/2.3﴿ + ﴾1/2.3 ‐ 1/3.4﴿ +...+ ﴾1/98.99 ‐ 1/99.100﴿
= 1/1.2 ‐ 1/2.3 + 1/2.3 ‐ 1/3.4 + ... + 1/98.99 ‐ 1/99.100
= 1/1.2 ‐ 1/99.100
= 1/2 ‐ 1/9900
= 4950/9900 ‐ 1/9900
= 4949/9900.
Vậy A = 4949 / 9900

Bn làm sai r . kết quả là \(\frac{101}{297}\) nhưng mik ko bt cách giải thôi

xin lỗi nha,gửi lời giải nhầm người

\(A=\left(1-\frac{2}{2\cdot3}\right)\cdot\left(1-\frac{2}{3\cdot4}\right)\cdot\left(1-\frac{2}{4\cdot5}\right)\cdot...\cdot1-\frac{2}{99\cdot100}\)

\(2A=1-\left(\frac{1}{2\cdot3}\cdot\frac{1}{3\cdot4}\cdot\frac{1}{4\cdot5}\cdot...\cdot\frac{1}{99\cdot100}\right)\)

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

\(2A=1-\left(\frac{1}{2}-\frac{1}{100}\right)\)

\(2A=1-\frac{49}{100}\)

\(2A=\frac{51}{100}\)

\(A=\frac{51}{100}:2\)

\(A=\frac{51}{200}\)

\(\left(1-\frac{2}{2.3}\right)\left(1-\frac{2}{3.4}\right)\left(1-\frac{2}{4.5}\right)...\left(1-\frac{2}{99.100}\right)\)

\(=\frac{4}{2.3}.\frac{10}{3.4}.\frac{18}{4.5}...\frac{9898}{99.100}\)

\(=\frac{1.4}{2.3}.\frac{2.5}{3.4}.\frac{3.6}{4.5}...\frac{98.101}{99.100}\)

\(=\frac{1.2.3...98}{2.3.4...99}.\frac{4.5.6...101}{3.4.5..100}\)

\(=\frac{1}{99}.\frac{101}{3}=\frac{101}{297}\)

\(=2\left(\frac{1}{2}-\frac{1}{2.3}\right).2\left(\frac{1}{2}-\frac{1}{3.4}\right)...2\left(\frac{1}{2}-\frac{2}{99.100}\right)\)
\(=2^{89}.\left(\frac{1}{2}.98-\frac{1}{2}+\frac{1}{100}\right)\)

\(=2^{98}.\left(49-\frac{49}{100}\right)\)

= \(\frac{2^{98}.4851}{100}\)

=\(2\left(\frac{1}{2}-\frac{1}{2.3}\right).2\left(\frac{1}{2}-\frac{1}{3.4}\right)....2\left(\frac{1}{2}-\frac{1}{99.100}\right)\)

=\(2^{89}\left(\frac{1}{2}.98-\frac{1}{2}+\frac{1}{100}\right)\)

\(=2^{98}.\left(49-\frac{49}{100}\right)=\frac{2^{98}.4851}{100}\)

Ta thấy: \(1-\frac{2}{n.\left(n+1\right)}=\frac{n.\left(n+1\right)-2}{n.\left(n+1\right)}=\frac{n^2+n-1-1}{n.\left(n+1\right)}=\frac{\left(n^2-1\right)+\left(n-1\right)}{n.\left(n+1\right)}=\frac{\left(n-1\right).\left(n+1\right)+\left(n-1\right)}{n.\left(n+1\right)}=\frac{\left(n-1\right).\left(n+2\right)}{n.\left(n+1\right)}\)

Lại có: \(\left(1-\frac{2}{2.3}\right).\left(1-\frac{2}{3.4}\right).\left(1-\frac{2}{4.5}\right).....\left(1-\frac{2}{99.100}\right)\)

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

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

\(=\frac{1.4}{2.3}.\frac{2.5}{3.4}.\frac{3.6}{4.5}.....\frac{98.101}{99.100}\)

\(=\frac{1.4.2.5.3.6.....98.101}{2.3.3.4.4.5.....99.100}\)

\(=\frac{\left(1.2.3.....98\right).\left(4.5.6.....101\right)}{\left(2.3.4.....99\right).\left(3.4.5.....100\right)}\)

\(=\frac{1.101}{99.3}\)

\(=\frac{101}{297}\)

\(1-\frac{2}{2}.3\) hay là \(1-\frac{2}{2.3}\)

Ta có:  \(1-\frac{2}{n.\left(n+1\right)}\)

          =\(\frac{n.\left(n+1\right)-2}{n\left(n+1\right)}\)

          =\(\frac{n^2+n-2}{n.\left(n+1\right)}\) 

          =\(\frac{\left(n^2-1\right)+\left(n-1\right)}{n.\left(n+1\right)}\)

          =\(\frac{\left(n-1\right).\left(n+1\right)+\left(n-1\right)}{n.\left(n+1\right)}\) 

          =\(\frac{\left(n-1\right).\left(n+1+1\right)}{n.\left(n+1\right)}\)

          =\(\frac{\left(n-1\right).\left(n+2\right)}{n.\left(n+1\right)}\)

=>\(1-\frac{2}{n.\left(n+1\right)}=\frac{\left(n-1\right).\left(n+2\right)}{n.\left(n+1\right)}\left(1\right)\)

Lại có: \(M=\left(1-\frac{2}{2.3}\right).\left(1-\frac{2}{3.4}\right).\left(1-\frac{2}{4.5}\right)....\left(1-\frac{2}{99.100}\right)\)

=>      \(M=\left(1-\frac{2}{2.\left(2+1\right)}\right).\left(1-\frac{2}{3.\left(3+1\right)}\right).\left(1-\frac{2}{4.\left(4+1\right)}\right)....\left(1-\frac{2}{99.\left(99+1\right)}\right)\left(2\right)\)

Thay (1) vào (2) ta được:

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

=>     \(M=\frac{1.4}{2.3}.\frac{2.5}{3.4}.\frac{3.6}{4.5}....\frac{98.101}{99.100}\)

=>     \(M=\frac{1.4.2.5.3.6....98.101}{2.3.3.4.4.5....99.100}\)

=>     \(M=\frac{\left(1.2.3....98\right).\left(4.5.6....101\right)}{\left(2.3.4....99\right).\left(3.4.5....100\right)}\)

=>     \(M=\frac{1.101}{99.3}\)

=> \(M=\frac{101}{297}\)

Vậy \(M=\frac{101}{297}\)