Question

\(\dfrac{1}{1.2}+\dfrac{1}{1.2.3}+...+\dfrac{1}{1.2.3...2021}\)

Answer 1

`2x^2 +2xy - 4y^2`

`= 2(x^2 + xy - 2y^2)`

`= 2( (x^2 -xy) + (2xy - 2y^2))`

`= 2( x(x -y) + 2y(x - y))`

`= 2 (x+2y)(x -y) `

`3x^2 +8xy - 3y^2`

`= (3x^2 - xy) + (9xy - 3y^2)`

`= x(3x-y) + 3y(3x - y)`

`= (x+3y)(3x-y)`

`x^2 - x -xy -2y^2 +2y`

`= x^2 - x + xy - 2xy -2y^2 +2y`

`= (x^2 - 2xy) + (2y - x) + (xy - 2y^2)`

`=  x(x - 2y) - (x - 2y) + y(x - 2y)`

`= (x-1+y)(x - 2y)`

Answer 2

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