Đáp án + Giải thích các bước giải:
1)
`\frac{2}{2.3} + \frac{2}{3.4} + \frac{2}{4.5} +...+ \frac{2}{99.100}`
`=\frac{2}{1}\times(\frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} +...+ \frac{1}{99} - \frac{1}{100})`
`=2 \times (\frac{1}{2} - \frac{1}{100})`
`=2 \times (\frac{50}{100} - \frac{1}{100})`
`=2 \times \frac{49}{100}`
`=\frac{39}{50}`
2)
`\frac{6}{1.3} + \frac{6}{3.5} + \frac{6}{5.7} +...+ \frac{6}{49.51}`
`=\frac{6}{2}\times ( 1 - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} +...+ \frac{1}{49} -\frac{1}{51})`
`=3 \times (1 - \frac{1}{51})`
`=3 \times \frac{50}{51}`
`=\frac{50}{17}`
3)
`\frac{2}{1.3} + \frac{2}{3.5} + \frac{2}{5.7} +...+ \frac{2}{2009.2011}`
`=1 - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} +...+ \frac{1}{2009} - \frac{1}{2011}`
`=1 - \frac{1}{2011}`
`= \frac{2010}{2011}`
Vì `\frac{2010}{2011} < 1 \Rightarrow \frac{2}{1.3} + \frac{2}{3.5} + \frac{2}{5.7} +...+ \frac{2}{2009.2011} < 1`
