Question

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Answer 1

$\frac{2}{1 \times 2} + \frac{2}{2 \times 3} + . . . + \frac{2}{19 \times 20} = 2 \times (\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + . . . + \frac{1}{19 \times 20}) = 2 \times (\frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + . . . + \frac{1}{19} - \frac{1}{20}) = 2 \times (1 - \frac{1}{20}) = 2 \times \frac{19}{20} = \frac{19}{10}$ ..

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Answer 2

$\frac{2}{1 \times 2} + \frac{2}{2 \times 3} + . . . + \frac{2}{19 \times 20} = 2 \times (\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + . . . + \frac{1}{19 \times 20}) = 2 \times (\frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + . . . + \frac{1}{19} - \frac{1}{20}) = 2 \times (1 - \frac{1}{20}) = 2 \times \frac{19}{20} = \frac{19}{10}$ ..

Answer 3

`2/(1xx2)+2/(2xx3)+2/(3xx4)+...+2/(18xx19)+2/(19xx20)`

`=2xx(1/(1xx2)+1/(2xx3)+1/(3xx4)+...+1/(18xx19)+1/(19xx20))`

`=2xx(1/1-1/2+1/2-1/3+1/3-1/4+...+1/18-1/19+1/19-1/20)`

`=2xx[1/1-(1/2-1/2)-(1/3-1/3)-(1/4-1/4)-...-(1/18-1/18)-(1/19-1/19)-1/20)]`

`=2xx(1-1/20)`

`=2xx(20/20-1/20)`

`=2xx19/20`

`=38/20`

`=19/10`

Answer 4

Đặt A=21x2+22x3+23x4+...+218x19+219x20 $A = \frac{2}{1 x 2} + \frac{2}{2 x 3} + \frac{2}{3 x 4} + . . . + \frac{2}{18 x 19} + \frac{2}{19 x 20}$

A2=11x2+12x3+13x4+...+118x19+119x20 $\frac{A}{2} = \frac{1}{1 x 2} + \frac{1}{2 x 3} + \frac{1}{3 x 4} + . . . + \frac{1}{18 x 19} + \frac{1}{19 x 20}$

A2=2−11x2+3−22x3+4−33x4+...+19−1818x19+20−1919x20 $\frac{A}{2} = \frac{2 - 1}{1 x 2} + \frac{3 - 2}{2 x 3} + \frac{4 - 3}{3 x 4} + . . . + \frac{19 - 18}{18 x 19} + \frac{20 - 19}{19 x 20}$

A2=1−12+12−13+13−14+...+118−119+119−120=1−120=1920 $\frac{A}{2} = 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + . . . + \frac{1}{18} - \frac{1}{19} + \frac{1}{19} - \frac{1}{20} = 1 - \frac{1}{20} = \frac{19}{20}$

A=2x1920=1910

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Answer 5

Đặt A=21x2+22x3+23x4+...+218x19+219x20 $A = \frac{2}{1 x 2} + \frac{2}{2 x 3} + \frac{2}{3 x 4} + . . . + \frac{2}{18 x 19} + \frac{2}{19 x 20}$

A2=11x2+12x3+13x4+...+118x19+119x20 $\frac{A}{2} = \frac{1}{1 x 2} + \frac{1}{2 x 3} + \frac{1}{3 x 4} + . . . + \frac{1}{18 x 19} + \frac{1}{19 x 20}$

A2=2−11x2+3−22x3+4−33x4+...+19−1818x19+20−1919x20 $\frac{A}{2} = \frac{2 - 1}{1 x 2} + \frac{3 - 2}{2 x 3} + \frac{4 - 3}{3 x 4} + . . . + \frac{19 - 18}{18 x 19} + \frac{20 - 19}{19 x 20}$

A2=1−12+12−13+13−14+...+118−119+119−120=1−120=1920 $\frac{A}{2} = 1 - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + . . . + \frac{1}{18} - \frac{1}{19} + \frac{1}{19} - \frac{1}{20} = 1 - \frac{1}{20} = \frac{19}{20}$

A=2x1920=1910

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