1. Ta chứng minh bất đẳng thức phụ dưới đây: \(\frac{1}{\sqrt{x}\left(x+1\right)}=\frac{\sqrt{x}}{x\left(x+1\right)}=\sqrt{x}\left(\frac{1}{x}-\frac{1}{x+1}\right)=\sqrt{x}\left(\frac{1}{\sqrt{x}}-\frac{1}{\sqrt{x+1}}\right)\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{x+1}}\right)\)\(=\left(1+\frac{\sqrt{x}}{\sqrt{x+1}}\right)\left(\frac{1}{\sqrt{x}}-\frac{1}{\sqrt{x+1}}\right)< 2\left(\frac{1}{\sqrt{x}}-\frac{1}{\sqrt{x+1}}\right)\)

Áp dụng  : \(\frac{1}{\sqrt{1}.2}< 2.\left(1-\frac{1}{\sqrt{2}}\right)\)

\(\frac{1}{\sqrt{2}.3}< 2.\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}\right)\)

...................................

\(\frac{1}{\sqrt{2015}.2016}< 2.\left(\frac{1}{\sqrt{2015}}-\frac{1}{\sqrt{2016}}\right)\)

Cộng các BĐT trên với nhau được : \(\frac{1}{2}+\frac{1}{3\sqrt{2}}+\frac{1}{4\sqrt{3}}+...+\frac{1}{2016\sqrt{2015}}< 2\left(1-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{2015}}-\frac{1}{\sqrt{2016}}\right)=2\left(1-\frac{1}{\sqrt{2016}}\right)< 2\left(1-\frac{1}{\sqrt{2025}}\right)=\frac{88}{45}\)

Từ đó suy ra đpcm

Cái ............... là gì vậy bn

   Đặt \(A=\frac{a^2}{a^2-1}-\frac{a^2}{1+a^2}.\left(\frac{a}{a+1}+\frac{1}{a^2+a}\right)\)

Ta có:\(A=\frac{a^2}{a^2-1}-\frac{a^2}{1+a^2}.\frac{a}{a+1}-\frac{a^2}{1+a^2}.\frac{1}{a^2+a}\)

          \(A=\frac{a^2}{a^2-1}-\frac{a^3}{a+a^3+a^2+1}-\frac{a^2}{a+a^2+a^3+a^4}\)

A= \(\frac{1,11+0,19-1,3.2}{2,06+0,54}-\left(\frac{1}{2}+\frac{1}{3}\right):2=\frac{-\frac{131}{100}}{\frac{13}{5}}-\frac{5}{6}:2\)

=\(-\frac{131}{260}-\frac{5}{12}=-\frac{359}{390}\)

B= \(\left(5\frac{7}{8}-2\frac{1}{4}-0,5\right):2\frac{23}{26}=\left(\frac{47}{8}-\frac{9}{4}-\frac{1}{2}\right):\frac{75}{26}=\frac{25}{8}.\frac{26}{75}=\frac{13}{12}\)

b) ta có : A=\(-\frac{359}{390}\approx-0,9\)

B= \(\frac{13}{12}\approx1,08\)

=> A<x<B mà x nguyên => x=0 hoặc x=1

Sai rồi

\(\left[\frac{x}{\left(x+4\right)\left(x-4\right)}-\frac{x-4}{x\left(x+4\right)}\right]:\frac{2\left(x-2\right)}{x\left(x+4\right)}\)\(=\left[\frac{x^2-\left(x-4\right)^2}{x\left(x+4\right)\left(x-4\right)}\right].\left[\frac{x\left(x+4\right)}{2\left(x-2\right)}\right]\)\(=\left(\frac{x^2-x^2+8x-16}{x\left(x+4\right)\left(X-4\right)}\right).\frac{x\left(x+4\right)}{2\left(x-2\right)}=\frac{8\left(x-2\right).x\left(x+4\right)}{x\left(x+4\right)\left(x-4\right).2\left(x-2\right)}=\frac{4}{x-4}\)

2.ta có |x-1|+(y+2)mũ 20=0=>x-1=0 đồng thời y+2=0

<=>x=1 và y=-2

Thay x=1 y=-2 vào B ta có:13.(1)^5-5.(-2)^3+2016=1989

\(C=\frac{x}{x-3}-\frac{x^2+3x}{2x+3}\left(\frac{x+3}{x^2-3x}-\frac{x}{x^2-9}\right)\)

=>\(C=\frac{x}{x-3}-\frac{x\left(x+3\right)}{2x+3}.\left[\frac{x+3}{x\left(x-3\right)}-\frac{x}{\left(x-3\right)\left(x+3\right)}\right]\)

=>\(C=\frac{x}{x-3}-\frac{x\left(x+3\right)}{2x+3}\left[\frac{\left(x+3\right)^2}{x\left(x-3\right)\left(x+3\right)}-\frac{x^2}{x\left(x-3\right)\left(x+3\right)}\right]\)

=>\(C=\frac{x}{x-3}-\frac{x\left(x+3\right)}{2x+3}.\frac{\left(x+3\right)^2-x^2}{x\left(x-3\right)\left(x+3\right)}\)

=>\(C=\frac{x}{x-3}-\frac{x\left(x+3\right)}{2x+3}.\frac{\left(x+3-x\right)\left(x+3+x\right)}{x\left(x-3\right)\left(x+3\right)}\)

=>\(C=\frac{x}{x-3}-\frac{x\left(x+3\right)}{2x+3}.\frac{3\left(2x+3\right)}{x\left(x-3\right)\left(x+3\right)}\)

=>\(C=\frac{x}{x-3}-\frac{3}{x-3}\)

=>\(C=\frac{x-3}{x-3}\)

=>C=1

\(A=\frac{x\left|x-2\right|}{x^2+8x-20}=\frac{x\left|x-2\right|}{x^2-2x+10x-20}=\frac{x\left|x-2\right|}{x\left(x-2\right)+10\left(x-2\right)}=\frac{x\left|x-2\right|}{\left(x+10\right)\left(x-2\right)}\)

Xét \(x-2\ge0\Leftrightarrow x\ge2\) ta có :

\(A=\frac{x\left(x-2\right)}{\left(x+10\right)\left(x-2\right)}=\frac{x}{x+10}\)

Xét \(x-2< 0\Leftrightarrow x< 2\) ta có :

\(A=\frac{x\left(2-x\right)}{\left(x+10\right)\left(x-2\right)}=\frac{-x}{x+10}\)

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