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Xét vế trái : \(\left(\sqrt{n+1}-\sqrt{n}\right)^2=2n+1-2\sqrt{n}.\sqrt{n+1}\)
Xét vế phải : \(\sqrt{\left(2n+1\right)^2}-\sqrt{\left(2n+1\right)^2-1}=\left|2n+1\right|-\sqrt{\left(2n+1-1\right)\left(2n+1+1\right)}=2n+1-2\sqrt{n}.\sqrt{n+1}\)
=> VT = VP
=> đpcm
1. Ta có: \(x+y+z=\sqrt{xy}+\sqrt{yz}+\sqrt{zx}\)
\(\Rightarrow\left(x+y+z\right)^2=\left(\sqrt{xy}+\sqrt{yz}+\sqrt{zx}\right)^2\)
\(\Leftrightarrow x^2+y^2+z^2+2xy+2yz+2xz=xy+yz+zx+2y\sqrt{xz}+2z\sqrt{xy}+2x\sqrt{yz}\)
\(\Leftrightarrow x^2+y^2+z^2+xy+yz+zx-2y\sqrt{xz}-2z\sqrt{xy}-2x\sqrt{yz}=0\)
\(\Leftrightarrow\left(x-\sqrt{yz}\right)^2+\left(y-\sqrt{xz}\right)^2+\left(z-\sqrt{xy}\right)^2=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=\sqrt{yz}\\y=\sqrt{xz}\\z=\sqrt{xy}\end{matrix}\right.\)
\(\Rightarrow x^2+y^2+z^2-xy-yz-zx=0\)
\(\Leftrightarrow2x^2+2y^2+2z^2-2xy-2yz-2zx=0\)
\(\Leftrightarrow\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2=0\Rightarrow x=y=z\)
Bài 1:
\(x+y+z=\sqrt{xy}+\sqrt{yz}+\sqrt{xz}\)
\(\Leftrightarrow x+y+z-\sqrt{xy}-\sqrt{yz}-\sqrt{xz}=0\)
\(\Leftrightarrow 2x+2y+2z-2\sqrt{xy}-2\sqrt{yz}-2\sqrt{xz}=0\)
\(\Leftrightarrow (x+y-2\sqrt{xy})+(y+z-2\sqrt{yz})+(z+x-2\sqrt{xz})=0\)
\(\Leftrightarrow (\sqrt{x}-\sqrt{y})^2+(\sqrt{y}-\sqrt{z})^2+(\sqrt{z}-\sqrt{x})^2=0\)
Vì \( (\sqrt{x}-\sqrt{y})^2;(\sqrt{y}-\sqrt{z})^2;(\sqrt{z}-\sqrt{x})^2\geq 0, \forall x,y,z>0\) nên để tổng của chúng bằng $0$ thì:
\( (\sqrt{x}-\sqrt{y})^2=(\sqrt{y}-\sqrt{z})^2=(\sqrt{z}-\sqrt{x})^2=0\)
\(\Rightarrow x=y=z\) (đpcm)
Xét vế trái : \(\left(\sqrt{n+1}-\sqrt{n}\right)^2=2n+1-2\sqrt{n}.\sqrt{n+1}\)
Xét vế phải : \(\sqrt{\left(2n+1\right)^2}-\sqrt{\left(2n+1\right)^2-1}=\left|2n+1\right|-\sqrt{\left(2n+1-1\right)\left(2n+1+1\right)}\)
\(=2n+1-\sqrt{2n.2\left(n+1\right)}=2n+1-2\sqrt{n}.\sqrt{n+1}\)
=> VT = VP => đpcm
A.\(\frac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{\left(n+1\right)^2n-n^2\left(n+1\right)}\) \(=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{n\left(n+1\right)\left(n+1-n\right)}=\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{n\left(n+1\right)}\)
=\(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\)
b. ap dungtinh B =\(\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{99}}-\frac{1}{\sqrt{100}}=1-\frac{1}{10}=\frac{9}{10}\)
Câu trả lời ở đây: https://dethihsg.com/de-thi-hoc-sinh-gioi-toan-9-phong-gddt-cam-thuy-2011-2012/amp/
Ta có :
\(\sqrt{\left(n+1\right)^2}+\sqrt{n^2}=\left|n+1\right|+\left|n\right|=\frac{\left[\left|n+1\right|+\left|n\right|\right]\left[\left|n+1\right|-\left|n\right|\right]}{\left|n+1\right|-\left|n\right|}\)
\(=\frac{\left|n+1\right|^2-\left|n\right|^2}{\left|n+1\right|-\left|n\right|}=\frac{\left(n+1\right)^2-n^2}{\left(n+1\right)-n}=\left(n+1\right)^2-n^2\)(đpcm)