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23 tháng 3 2017

\(\frac{a^{2009}-b^{2009}}{a^{2009}+b^{2009}}>\frac{a^{2008}-b^{2008}}{a^{2008}+b^{2008}}\)

\(\Leftrightarrow\left(a^{2009}-b^{2009}\right)\left(a^{2008}+b^{2008}\right)-\left(a^{2009}+b^{2009}\right)\left(a^{2008}-b^{2008}\right)>0\)

\(\Leftrightarrow a^{2009}b^{2008}-b^{2009}a^{2008}>0\)

\(\Leftrightarrow a^{2008}b^{2008}\left(a-b\right)>0\)(đúng vì \(a>b>0\))

Vậy \(\frac{a^{2009}-b^{2009}}{a^{2009}+b^{2009}}>\frac{a^{2008}-b^{2008}}{a^{2008}+b^{2008}}\)

23 tháng 3 2017

\(\frac{a^{2009}-b^{2009}}{a^{2009}+b^{2009}}>\frac{a^{2008}-b^{2008}}{a^{2008}+b^{2008}}\)

Bài 1: 

Ta có: \(a+b\ge2\sqrt{ab}\)

\(b+c\ge2\sqrt{bc}\)

\(a+c\ge2\sqrt{ac}\)

Do đó: \(2\left(a+b+c\right)\ge2\left(\sqrt{ab}+\sqrt{bc}+\sqrt{ac}\right)\)

hay \(a+b+c\ge\sqrt{ab}+\sqrt{cb}+\sqrt{ac}\)

NV
30 tháng 9 2020

Do \(\left\{{}\begin{matrix}a^{2008}\ge0\\b^{2008}\ge0\\c^{2008}\ge0\\a^{2008}+b^{2008}+c^{2008}=1\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a^{2008}\le1\\b^{2008}\le1\\c^{2008}\le1\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}\left|a\right|\le1\\\left|b\right|\le1\\\left|c\right|\le1\end{matrix}\right.\)

\(\Rightarrow a^{2009}+b^{2009}+c^{2009}\le a^{2008}+b^{2008}+c^{2008}\)

\(\Rightarrow a^{2009}+b^{2009}+c^{2009}\le1\)

Dấu "=" xảy ra khi và chỉ khi \(\left(a;b;c\right)=\left(0;0;1\right)\) và hoán vị

Khi đó \(a^{2007}+b^{2008}+c^{2009}+2020=1+2020=2021\)

26 tháng 10 2019

đặt \(2008=a\)

\(\sqrt{1+a^2+\frac{a^2}{\left(a+1\right)^2}}=\sqrt{\left(a+1\right)^2-\frac{2\left(a+1\right).a}{a+1}+\left(\frac{a}{a+1}\right)^2}=\)\(\sqrt{\left(a+1-\frac{a}{a+1}\right)^2}=a+1-\frac{a}{a+1}\)=2008+1- \(\frac{2008}{2009}\)

=> A= 2008+1 = 2009

26 tháng 9 2021

Đặt \(2008=a\)

\(\Leftrightarrow A=\sqrt{1+a^2+\dfrac{a^2}{\left(a+1\right)^2}}+\dfrac{a}{a+1}\\ A=\sqrt{\left(a+1\right)^2-\dfrac{2a\left(a+1\right)}{a+1}+\dfrac{a^2}{\left(a+1\right)^2}}+\dfrac{a}{a+1}\\ A=\sqrt{\left(a+1-\dfrac{a}{a+1}\right)^2}+\dfrac{a}{a+1}\\ A=a+1-\dfrac{a}{a+1}+\dfrac{a}{a+1}=a+1=2009\left(đpcm\right)\)

31 tháng 7 2018

\(A=\sqrt{1+2008^2+\dfrac{2008^2}{2009^2}}+\dfrac{2008}{2009}=\sqrt{2008^2+2.2008+1-2.2008+\dfrac{2008^2}{2009^2}}+\dfrac{2008}{2009}=\sqrt{2009^2-2.2009.\dfrac{2008}{2009}+\dfrac{2008^2}{2009^2}}+\dfrac{2008}{2009}=\sqrt{\left(2009-\dfrac{2008}{2009}\right)^2}+\dfrac{2008}{2009}=2009\)

Vậy , A có giá trị là số nguyên .

2 tháng 11 2018

\(B=\sqrt{1+2008^2+\dfrac{2008^2}{2009^2}}+\dfrac{2008}{2009}=\sqrt{\dfrac{2009^2+2008^2.2009^2+2008^2}{2009^2}}+\dfrac{2008}{2009}=\dfrac{\sqrt{2009^2+\left(2009-1\right)^2.2009^2+2008^2}}{2009}+\dfrac{2008}{2009}=\dfrac{\sqrt{2009^2+2009^4-2.2009.2009^2+2009^2+2008^2}+2008}{2009}=\dfrac{\sqrt{2009^4+2.2009^2-2.\left(2008+1\right).2009^2+2008^2}+2008}{2009}=\dfrac{\sqrt{2009^4+2.2009^2-2.2008.2009^2-2.2009^2+2008^2}+2008}{2009}=\dfrac{\sqrt{2009^4-2.2008.2009^2+2008^2}+2008}{2009}=\dfrac{\sqrt{\left(2009^2-2008\right)^2}+2008}{2009}=\dfrac{2009^2-2008+2008}{2009}=2009\in N\)

Vậy B có giá trị là một số tự nhiên

3 tháng 10 2019

Xét các số thực a, b, c thỏa mãn \(a+b+c=0\)

\(\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}=\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{2}{ab}+\frac{2}{bc}+\frac{2}{ca}-\frac{2}{ab}-\frac{2}{bc}-\frac{2}{ca}}\)

\(=\sqrt{\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2-2.\frac{a+b+c}{abc}}\)

\(=\sqrt{\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2}=\left|\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right|\)

Ta có:

\(B=\sqrt{1+2008^2+\frac{2008^2}{2009^2}}+\frac{2008}{2009}\)

\(=\sqrt{2008^2}.\sqrt{\frac{1}{2018^2}+\frac{1}{1^2}+\frac{1}{2009^2}}+\frac{2008}{2009}\)

\(=2008.\sqrt{\frac{1}{2018^2}+\frac{1}{1^2}+\frac{1}{\left(-2009\right)^2}}+\frac{2008}{2009}\)

\(=2008.\left|\frac{1}{2008}+1-\frac{1}{2009}\right|+\frac{2008}{2009}\)

\(=2008.\left(\frac{1}{2008}+1-\frac{1}{2009}\right)+\frac{2008}{2009}\)

\(=2008.\left(\frac{1}{2008}+1-\frac{1}{2009}+\frac{1}{2009}\right)\)

\(=2008.\frac{2009}{2008}=2009\in\text{N}\)