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

\(=\frac{2^2}{1\cdot3}\cdot\frac{3^2}{2\cdot4}\cdot\frac{4^2}{3\cdot5}\cdot...\cdot\frac{99^2}{98\cdot100}\cdot\frac{100^2}{99\cdot101}=\frac{200}{101}\)

\(A=\left(1+\frac{1}{1.3}\right)\left(1+\frac{1}{2.4}\right)\left(1+\frac{1}{3.5}\right)......\left(1+\frac{1}{99.100}\right)\)

\(=\left(1+\frac{1}{2^2-1}\right)\left(1+\frac{1}{3^2-1}\right)......\left(1+\frac{1}{100^2-1}\right)\)

\(=\frac{2^2}{1.3}.\frac{3^2}{2.4}..............\frac{100^2}{99.100}=\frac{200}{101}\)

T nha

\(S=\frac{1}{2}.\left(\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-...-\frac{1}{99}+\frac{1}{99}-\frac{1}{101}\right)\)

\(S=\frac{1}{2}.\left(1-\frac{1}{101}\right)=\frac{1}{2}\times\frac{100}{101}=\frac{50}{101}\)

\(S=\frac{1}{1.3}+\frac{1}{3.5}+...+\frac{1}{99.100}\)

\(S=1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{99}-\frac{1}{100}\)

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

\(S=\frac{99}{100}\)

\(S=\frac{1}{1\times3}+\frac{1}{3\times5}+...+\frac{1}{99\times101}\) chứ bạn 

\(S=\left(\dfrac{1}{1\cdot3}+\dfrac{1}{3\cdot5}+...+\dfrac{1}{97\cdot99}\right)+\left(\dfrac{1}{2\cdot4}+\dfrac{1}{4\cdot6}+...+\dfrac{1}{98\cdot100}\right)-\dfrac{49}{99}\)

\(=\dfrac{1}{2}\left(\dfrac{2}{1\cdot3}+\dfrac{2}{3\cdot5}+...+\dfrac{2}{97\cdot99}\right)+\dfrac{1}{2}\left(\dfrac{2}{2\cdot4}+\dfrac{2}{4\cdot6}+...+\dfrac{2}{98\cdot100}\right)-\dfrac{49}{99}\)

\(=\dfrac{1}{2}\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{97}-\dfrac{1}{99}\right)+\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+...+\dfrac{1}{98}-\dfrac{1}{100}\right)-\dfrac{49}{99}\)

\(=\dfrac{1}{2}\cdot\dfrac{98}{99}+\dfrac{1}{2}\cdot\dfrac{49}{100}-\dfrac{49}{99}\)

\(=\dfrac{49}{200}\)

Đề như thế nào ý

2s=2/

4.9 hay 4.6

S=(1/1.3+1/3.5+.....+1/7.9) + (1/2.4+1/4.6+....+1/8.10)

2S=1/2.(1-1/9+(1/2-1/10))

2S=1/2.(8/9 + 2/5)

2S=1/2.58/45

2S=29/45

S=29/45:2

S=29/90

S=(1/1.3+1/3.5+.....+1/7.9) + (1/2.4+1/4.6+....+1/8.10)

2S=1/2.(1-1/9+(1/2-1/10))

2S=1/2.(8/9 + 2/5)

2S=1/2.58/45

2S=29/45

S=29/45:2

S=29/90

S = (1/1.3 + 1/3.5 + ... + 1/7.9) + (1/2.4 + 1/4.6 + ... + 1/8.10)

S = 2(1/1 - 1/3 + 1/3 - 1/5 + ... + 1/7 - 1/9) + 1/2(1/2 - 1/4 + 1/4 - 1/6 + ... + 1/8 - 1/10) 

S = 2.(1 - 1/9) + 1/2.(1/2 - 1/10)

S = 16/9 + 1/5 = 89/45