CMR
\(\sqrt{1+2+3+..+\left[n-1\right]+\left[n-1\right]+...+3+2+1}\) =n
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a) \(\left|x+2\right|+\left|2y-1\right|=0\)
\(\Leftrightarrow\hept{\begin{cases}\left|x+2\right|=0\\\left|2y-1\right|=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x+2=0\\2y-1=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x=-2\\y=0,5\end{cases}}\)
Vậy (x; y) = (-2; 0,5)
b) \(\left|x-y\right|+\left|2x+3\right|=0\Leftrightarrow\hept{\begin{cases}\left|x-y\right|=0\\\left|2x+3\right|=0\end{cases}}\)
+) |2x + 3| = 0
2x + 3 = 0
2x = -3
x = -1,5
+) |x - y| = 0
x - y = 0
-1,5 - y = 0
y = -1,5
Vậy (x; y) = (-1,5; -1,5)
c, \(\left|2x+y\right|+\left|y+\left(1:4\right)\right|=0\)
\(\left|2x+y\right|+\left|y+\frac{1}{4}\right|=0\)
\(\Leftrightarrow\hept{\begin{cases}\left|2x+y\right|=0\\\left|y+\frac{1}{4}\right|=0\end{cases}}\)
\(\left|y+\frac{1}{4}\right|=0\Leftrightarrow y+\frac{1}{4}=0\Leftrightarrow y=-\frac{1}{4}\)
\(\left|2x+y\right|=0\Leftrightarrow2x+y=0\Leftrightarrow2x-\frac{1}{4}=0\Leftrightarrow2x=\frac{1}{4}\Leftrightarrow x=\frac{1}{8}\)
Vậy \(\left(x;y\right)=\left(\frac{1}{8};-\frac{1}{4}\right)\)
Ta thấy :
\(\left|x-2012\right|+\left|x-2013\right|=\left|x-2012\right|+\left|2013-x\right|\ge\left|x-2012+2013-x\right|=1\)
\(\Rightarrow VT\ge VP\)
Dấu "=" xảy ra \(\Leftrightarrow\left(x-2012\right)\left(2013-x\right)\ge0\Leftrightarrow2012\le x\le2013\)
Vậy \(2012\le x\le2013\)
\(b^2=ac\Rightarrow\frac{a}{b}=\frac{b}{c};c^2=bd\Rightarrow\frac{b}{c}=\frac{c}{d}\)
\(\Rightarrow\frac{a}{b}=\frac{b}{c}=\frac{c}{d}\)
Đặt \(\frac{a}{b}=\frac{b}{c}=\frac{c}{d}=k\)
\(\Rightarrow k^3=\frac{a^3}{b^3}=\frac{b^3}{c^3}=\frac{c^3}{d^3}=\frac{a^3+b^3+c^3}{b^3+c^3+d^3}\) (TC DTSBN) (1)
Ta lại có : \(k^3=\frac{a}{b}.\frac{a}{b}.\frac{a}{b}=\frac{a}{b}.\frac{b}{c}.\frac{c}{d}=\frac{a}{d}\) (2)
Từ (1) ; (2) \(\Rightarrow\frac{a^3+b^3+c^3}{b^3+c^3+d^3}=\frac{a}{d}\) (đpcm)
\(\sqrt{1+2+3+...+n-1+n-1+...+3+2+1}\)
\(=\sqrt{2\left[1+2+3+...+n-1\right]+n}\)
\(=\sqrt{\frac{2\left[n-1\right]n}{2}}+n=\sqrt{n^2}=n\)=> ĐPCM