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\(=\dfrac{1}{x}-\dfrac{1}{x+1}+\dfrac{1}{x+1}-\dfrac{1}{x+2}+...+\dfrac{1}{x+2013}-\dfrac{1}{x+2014}\)
=1/x-1/x+2014
\(=\dfrac{x+2014-x}{x\left(x+2014\right)}=\dfrac{2014}{x\left(x+2014\right)}\)
theo cách tính tổng (bn có thể xem lại ở toán 7 hay 6 j đấy) thì bt trên bằng 1/x - 1/(x+5)
từ đó tính tiếp nha bn
\(A=\frac{1}{x\left(x+1\right)}+\frac{1}{\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+.....+\frac{1}{\left(x+2013\right)\left(x+1014\right)}\)
\(\Leftrightarrow A=\frac{1}{x}-\frac{1}{x+1}+\frac{1}{x+1}-\frac{1}{x+2}+\frac{1}{x+2}-\frac{1}{x+3}+....+\frac{1}{x+2013}-\frac{1}{x+2014}\)
\(\Leftrightarrow A=\frac{1}{x}-\frac{1}{x+2014}\)
\(\Leftrightarrow A=\frac{x+2014-x}{x\left(x+2014\right)}=\frac{2014}{x\left(x+2014\right)}\)
Trả lời:
a, ( x + 1 )2 + ( x - 2 ) ( x + 3 ) - 4x
= x2 + 2x + 1 + x2 + 3x - 2x - 6 - 4x
= 2x2 - x - 5
b, ( x - 2 )2 + ( x + 1 )2 + 2 ( x - 2 ) ( - 1 - x )
= x2 - 4x - 4 + x2 + 2x + 1 + ( 2x - 4 ) ( - 1 - x )
= 2x2 - 2x - 3 - 2x - 2x2 + 4x + 4x
= 4x - 3
a) \(\left(x+1\right)^2+\left(x-2\right)\left(x+3\right)-4x\)
\(=\left(x^2+2x+1\right)+\left(x^2+x-6\right)-4x\)
\(=x^2+2x+1+x^2+x-6-4x\)
\(=2x^2-x-5\)
b) \(\left(x-2\right)^2+\left(x+1\right)^2+2\left(x-2\right)\left(-1-x\right)\)
\(=\left(x^2-4x+4\right)+\left(x^2+2x+1\right)+\left(2x-4\right)\left(-1-x\right)\)
\(=x^2-4x+4+x^2+2x+1+\left(-2x-2x^2+4+4x\right)\)
\(=x^2-4x+4+x^2+2x+1-2x-2x^2+4+4x\)
\(=9\)
Đặt biểu thức là A
\(\Rightarrow\)A=\(\dfrac{\left(x+1\right)-x}{x\left(x+1\right)}+\dfrac{\left(x+2\right)-\left(x+1\right)}{\left(x+1\right)\left(x+2\right)}+\dfrac{\left(x+3\right)-\left(x+2\right)}{\left(x+2\right)\left(x+3\right)}+...+\dfrac{\left(x+2014\right)-\left(x+2013\right)}{\left(x+2013\right)\left(x+2014\right)}\)
\(\Leftrightarrow\dfrac{x+1}{x\left(x+1\right)}-\dfrac{x}{x\left(x+1\right)}+\dfrac{x+2}{\left(x+1\right)\left(x+2\right)}-\dfrac{x+1}{\left(x+1\right)\left(x+2\right)}+...+\dfrac{x+2014}{\left(x+2013\right)\left(x+2014\right)}-\dfrac{x+2013}{\left(x+2013\right)\left(x+2014\right)}\)\(\Leftrightarrow\dfrac{1}{x}-\dfrac{1}{x+1}+\dfrac{1}{x+1}-\dfrac{1}{x+2}-\dfrac{1}{x+2}-...-\dfrac{1}{x+2013}+\dfrac{1}{x+2013}-\dfrac{1}{x+2014}.\)\(\Leftrightarrow\dfrac{1}{x}-\dfrac{1}{x+2014}\)
\(\Leftrightarrow\dfrac{x+2014-x}{x\left(x+2014\right)}\)
\(\dfrac{2014}{x\left(x+2014\right)}\)