Question

\(\frac{2009^{2008}-1}{2009^{2009}-1}< \frac{2009^{2007}+1}{2009^{2008}+1}\)

giúp mik lun nha

Answer 1

Trả lời

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Answer 2

Đặt \(A=\frac{2009^{2008}-1}{2009^{2009}-1}\)

\(\Rightarrow2009A=\frac{2009.\left(2009^{2008}-1\right)}{2009^{2009}-1}=\frac{2009^{2009}-2009}{2009^{2009}-1}\)

\(=\frac{2009^{2009}-1-2008}{2009^{2009}-1}=1-\frac{2008}{2009^{2009}-1}\)

Đặt \(B=\frac{2009^{2007}+1}{2009^{2008}+1}\)

\(\Rightarrow2009B=\frac{2009.\left(2009^{2007}+1\right)}{2009^{2008}+1}=\frac{2009^{2008}+2009}{2009^{2008}+1}\)

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

Vì : \(\frac{2008}{2009^{2009}-1}< \frac{2008}{2009^{2008}+1}\)

\(\Rightarrow A=1-\frac{2008}{2009^{2009}-1}< B=1+\frac{2008}{2009^{2008}+1}\)

Vậy \(\frac{2009^{2008}-1}{2009^{2009}-1}< \frac{2009^{2007}+1}{2009^{2008}+1}\)