`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)`
Chứng minh BĐT sau
a)\(\dfrac{1}{1.3}+\dfrac{1}{3.5}+...+\dfrac{1}{\left(2n-1\right)\left(2n+1\right)}< \dfrac{1}{2}\)
b)
\(\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+.....+\dfrac{1}{37.38.39}\)
\(\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+...+\dfrac{1}{48.49.50}\)
Tính nhanh tổng sau: \(\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+\dfrac{1}{3.4.5}+...+\dfrac{1}{10.11.12}\)
Cho S=\(\dfrac{5}{1.2.3}+\dfrac{8}{2.3.4}+\dfrac{11}{3.4.5}+...+\dfrac{6068}{2022.2023.2024}\)
So sánh S với 2
1. So sánh
a) \(A=\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{2^{2020}}+\dfrac{1}{2^{2021}}\) và B= \(\dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}+\dfrac{13}{60}\)
b) \(C=\dfrac{2019}{2021}+\dfrac{2021}{2022}\) và \(D=\dfrac{2020+2022}{2019+2021}.\dfrac{3}{2}\)
Tính : S = \(1-\dfrac{1}{2}+\dfrac{1}{3}-\)\(\dfrac{1}{4}+...+\dfrac{1}{2019}-\dfrac{1}{2020}+\dfrac{1}{2021}\)và
P = \(\dfrac{1}{2011}+\dfrac{1}{2012}+\dfrac{1}{2013}+...+\dfrac{1}{2020}+\dfrac{1}{2021}\)
Tính : \(\left(S-P\right)^{2022}\)
Tính nhanh:
1.( 1.2 ).( 1.2.3 )........( 1.2.3.....96 )
\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{2021^2}\). So sánh A và \(\dfrac{2020}{2021}\)
Cho a,b,c thỏa mãn a+b+c = 2021 và \(\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{c+a}=\dfrac{1}{2021}\)
Tính Q = \(\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\)
