Cho a>b>0.
CMR:
$\frac{a^{2015}-b^{2015}}{a^{2015}+b^{2015}}> \frac{a^{2014}-b^{2014}}{a^{2014}+b^{2014}}$
$\frac{a^{2015}-b^{2015}}{a^{2015}+b^{2015}}> \frac{a^{2014}-b^{2014}}{a^{2014}+b^{2014}}$
Started By vua thac mac, 20-07-2014 - 17:12
#2
Posted 20-07-2014 - 20:02
thuan192
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Sĩ quan
Cho a>b>0.
CMR:
$\frac{a^{2015}-b^{2015}}{a^{2015}+b^{2015}}> \frac{a^{2014}-b^{2014}}{a^{2014}+b^{2014}}$
$\Leftrightarrow 1-\frac{a^{2015}-b^{2015}}{a^{2015}+b^{2015}}< 1-\frac{a^{2014}-b^{2014}}{a^{2014}+b^{2014}}\Leftrightarrow \frac{b}{a^{2015}+b^{2015}}<\frac{1}{a^{2014}+b^{2014}}$
$\Leftrightarrow b.a^{2014}<a^{2015}$ (đúng vì $a>b$)
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Thuận
#3
Posted 20-07-2014 - 20:38
huyhoangfan
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Trung sĩ
Cho a>b>0.
CMR:
$\frac{a^{2015}-b^{2015}}{a^{2015}+b^{2015}}> \frac{a^{2014}-b^{2014}}{a^{2014}+b^{2014}}$
BĐT$\Leftrightarrow \frac{a^{2015}+b^{2015}-2b^{2015}}{a^{2015}+b^{2015}}> \frac{a^{2014}+b^{2014}-2b^{2014}}{a^{2014}+b^{2014}}\Leftrightarrow 1-\frac{2b^{2015}}{a^{2015}+b^{2015}}> 1-\frac{2b^{2014}}{a^{2014}+b^{2014}}\Leftrightarrow \frac{2b^{2015}}{a^{2015}+b^{2015}}< \frac{b^{2014}}{a^{2014}+b^{2014}}\Leftrightarrow \frac{1}{\frac{a^{2015}}{b^{2015}}+1}< \frac{1}{\frac{a^{2014}}{b^{2014}}+1}\Leftrightarrow \frac{a^{2015}}{b^{2015}}> \frac{a^{2014}}{b^{2014}}$. (vì $a>b$ )
BĐT cuối cùng đúng $\Rightarrow$ ĐPCM
Edited by huyhoangfan, 20-07-2014 - 20:56.
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