Question

Tính:

\(\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{99.100}\)

Answer 1

\(\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{99.100}\)

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

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

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

\(\frac{2}{1}.\frac{49}{100}\)

\(\frac{98}{100}=\frac{49}{50}\)

Answer 2

Đặt A = \(\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{99.100}\)

 A : 2 =  \(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{99.100}\)

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

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

 A : 2 = \(\frac{49}{100}\)

    A   = \(\frac{49}{50}\)

Answer 3

Đặt \(A=\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{99.100}\)

\(\Rightarrow A=2.\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{99.100}\right)\)

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

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

\(\Rightarrow A=2.\frac{49}{50}\)

\(\Rightarrow A=\frac{49}{25}\)

Answer 4

 A = 2/2*3 + 2/3*4 + 2/4*5 + ... + 2/99*100

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

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

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

2A = 1- 1/100 = 99/100

 A = 99/100 : 2 = 99/200

Answer 5

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

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

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

\(=2.\frac{49}{100}=\frac{49}{50}\)