[tex] 3^{2014}+ 4^{2014}< 5^{2014}<=> \\ <=> \frac{3^{2014}+ 4^{2014} }{5 ^{2014} }<1<=> \\ <=> (\frac{3}{5}) ^{2014} + (\frac{4}{5}) ^{2014}<1. [/tex] (Ramane de verificat daca este adevarat.)
Este evident ca [tex]( \frac{3}{5} ) ^{2014}< ( \frac{3}{5}) ^{2} = \frac{9}{25} [/tex] si [tex]( \frac{4}{5}) ^{2014}< (\frac{4}{5}) ^{2}= \frac{16}{25} .[/tex]
Deci [tex]( \frac{3}{5}) ^{2014} + (\frac{4}{5}) ^{2014} < \frac{9}{25}+ \frac{16}{25}= \frac{25}{25}=1, [/tex] deci este adevarat si prin urmare, rezulta ca [tex] 3^{2014}+ 4^{2014} < 5^{2014} .[/tex]
