\(A=\dfrac{4}{2.4}+\dfrac{4}{4.6}+\dfrac{4}{6.8}+...+\dfrac{4}{2008.2010}\)

\(=2\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{8}+...+\dfrac{1}{2008}-\dfrac{1}{2010}\right)\)

\(=2\left(\dfrac{1}{2}-\dfrac{1}{2010}\right)\)

\(=2.\dfrac{502}{1005}=\dfrac{1004}{1005}\)

Ta có:

\(\dfrac{a}{b}=\dfrac{c}{d}\)

\(\Rightarrow\dfrac{c}{a}=\dfrac{d}{b}\)

\(\dfrac{2a+c}{a}=\dfrac{2a}{a}+\dfrac{c}{a}=2+\dfrac{c}{a}\)

\(\dfrac{2b+d}{b}=\dfrac{2b}{b}+\dfrac{d}{b}=2+\dfrac{d}{b}=2+\dfrac{c}{a}\left(\text{Vì }\dfrac{d}{b}=\dfrac{c}{a}\right)\)

\(\Rightarrow\dfrac{2a+c}{a}=\dfrac{2b+d}{b}\)

\(\Rightarrow\dfrac{a}{2a+c}=\dfrac{b}{2b+d}\)

\(\RightarrowĐPCM\)

\(\dfrac{2x+4}{2015}-\dfrac{2x+4}{2016}=\dfrac{2x+4}{2017}-\dfrac{2x+4}{2018}\)

\(\Rightarrow\left(2x+4\right)\left(\dfrac{1}{2015}-\dfrac{1}{2016}\right)=\left(2x+4\right)\left(\dfrac{1}{2017}-\dfrac{1}{2018}\right)\)

Vì \(\dfrac{1}{2015}-\dfrac{1}{2016}\ne\dfrac{1}{2016}-\dfrac{1}{2017}\) nên \(2x+4=0\)

\(\Rightarrow2x=-4\)

\(\Rightarrow x=-2\)

Vậy, x = -2

\(\left(2x+5\right)+\left(4x+20\right)=5x+34\)

\(\left(2x+5\right)+\left(4x+20\right)=5x+34 \)

\(\Leftrightarrow2x+5+4x+20-5x-34=0\)

\(\Leftrightarrow x-9=0\)

\(\Leftrightarrow x=9\)

Vậy x=9

Gọi số học sinh giỏi của ba lớp 7A, 7B, 7C lần lượt là x, y, z.

Theo đề bài ta có : \(\dfrac{x}{2}=\dfrac{y}{4}=\dfrac{z}{6} \) và \(z-y=6\)

Áp dụng tính chất của dãy tỉ số bằng nhau ta có :

\(\dfrac{x}{2}=\dfrac{y}{4}=\dfrac{0z}{6}=\dfrac{z-y}{6-4}=\dfrac{6}{2}=3\)

\(\dfrac{x}{2}=3\Rightarrow x=3.2=6\)

\(\dfrac{y}{4}=3\Rightarrow y=3.4=12\)

\(\dfrac{z}{6}=3\Rightarrow z=3.6=18 \)

Vậy : 7A có 6 học sinh giỏi

7B có 12 học sinh giỏi

7C có 18 học sinh giỏi

a, \(5x^3-125=0\)

\(\Leftrightarrow5x^3=125\)

\(\Leftrightarrow x^3=25\)

\(\Leftrightarrow x=\sqrt[3]{25}\)

c, \(6x+13x+5=0\)

\(\Leftrightarrow19x+5=0\)

\(\Leftrightarrow19x=-5\)

\(\Leftrightarrow x=\dfrac{-5}{19}\)

\(P=\dfrac{1}{2}+\sqrt{x}\ge\dfrac{1}{2}\)

Dấu "=" xảy ra khi : \(x=0\)

\(Q=7-2\sqrt{x-1}\le7\)

Dấu "=" xảy ra khi : \(x=1\)

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