2/1 = 2
3/2 = 1,5
4/3 = 1,(3)
5/4 = 1,25
......
Observam ca valoarea fractiei scade pe masura ce sirul inainteaza.
Deci, 4567/4566 > 4568/4567, dar trebuie sa demonstram.
[tex] $Fie $k\geq 1 \\ \\ \dfrac{k+1}{k}\ \textgreater \ \dfrac{k+2}{k+1} \\ \\ \dfrac{(k+1)\cdot (k+1)}{k\cdot(k+1)}
\ \textgreater \ \dfrac{k\cdot (k+2)}{k\cdot (k+1)}\Big|\cdot \Big(k\cdot (k+1)\Big), \boxed{k\cdot (k+1) \rightarrow $ pozitiv $ }\\ \\ (k+1)(k+1)\ \textgreater \ k\cdot (k+2) \\ \\ (k+1)^2\ \textgreater \ k^2+2k \\ \\ k^2+2k+1\ \textgreater \ k^2+2k \\ \\ k^2+2k-k^2-2k+1\ \textgreater \ 0 \\ \\ 1\ \textgreater \ 0 \quad (A) \\ \\ \Rightarrow \dfrac{k+1}{k}\ \textgreater \ \dfrac{k+2}{k+1},\quad (\forall) $ $ k \geq 1 \\ \\ \\ \Rightarrow \boxed{\boxed{\dfrac{4567}{4566}\ \textgreater \ \dfrac{4568}{4567} }}[/tex]
