(2007+1)^(2007-1)*(2007-1)^(2007+1) ? 2007^(2*2007)

[Guest]

(2007+1)^(2007-1)*(2007-1)^(2007+1) ? 2007^(2*2007)
instead of ? put < or >

[shyamjayakannan]

let [tex]x=\frac{(2007+1)^{2007-1}(2007-1)^{2007+1}}{2007^{2\times2007}}\ldots(1)[/tex]

[tex]\Rightarrow x=\frac{\{(2007+1)(2007-1)\}^{2007-1}(2007-1)^2}{2007^{2\times2007}}=\frac{\left(2007^2-1\right)^{2007-1}(2007-1)^2}{2007^{2\times2007}}[/tex]

[tex]\Rightarrow x =\frac{\left(2007^2-1\right)^{2007-1}(2007-1)^2}{2007^{2\times2007}}\times\frac{2007^2-1}{2007^2-1}=\frac{\left(2007^2-1\right)^{2007}(2007-1)^2}{2007^{2\times2007}\left(2007^2-1\right)}=\left(\frac{2007^2-1}{2007^2}\right)^{2007}\times\frac{(2007-1)^2}{2007^2-1}[/tex]

[tex]\Rightarrow x=\left(1-\frac{1}{2007^2}\right)^{2007}\times\frac{(2007-1)^2}{(2007-1)(2007+1)}=\left(1-\frac{1}{2007^2}\right)^{2007}\times\frac{2007-1}{2007+1}[/tex]

So, [tex]x[/tex] is a product of two terms, both of which are less than [tex]1[/tex]. So [tex]x < 1\ldots(2)[/tex]

From [tex](1)[/tex] and [tex](2)[/tex], we get [tex]\boxed{(2007+1)^{2007-1}(2007-1)^{2007+1}<2007^{2\times2007}}[/tex]