Državno natjecanje 2004 SŠ4 3
Dodao/la:
arhiva1. travnja 2012. Nizovi realnih brojeva
![(x_n), (y_n), (z_n), n \in \mathbb{N}](/media/m/1/e/3/1e3d148d0240eaa3b3b7dc47c3b405b0.png)
, definirani su formulama
![x_{n+1}=\frac{2x_n}{x_n^2 - 1}\text{,}\,\,\, y_{n+1}=\frac{2y_n}{y_n^2 - 1}\text{,}\,\,\, z_{n+1}=\frac{2z_n}{z_n^2 - 1}\text{,}](/media/m/7/d/d/7dd85c953f051793740f9c1cc23c03f9.png)
a početni članovi su
![x_1 = 2](/media/m/f/e/8/fe8f9afc51e9f1ca8fd4457ae0d700a2.png)
,
![y_1 = 4](/media/m/4/d/e/4de084dd2f8865e345906bba34b9b05b.png)
i
![z_1](/media/m/1/2/c/12c460682a9669b12ec64d645af1bb87.png)
takav da vrijedi
![x_1y_1z_1 = x_1 + y_1 + z_1](/media/m/5/3/0/530a0534a3ed0dbfb133c690f01351ea.png)
.
![a)](/media/m/f/0/8/f0844437a160b45486aedcc02b92949d.png)
Provjerite da su za svaki
![n \in \mathbb{N}](/media/m/2/b/a/2ba27c6141ca415bb86bae1d237f1fac.png)
zadovoljeni uvjeti:
![x_n^2 \not= 1](/media/m/6/0/e/60e751923e8ff9760fc068c626bca4a4.png)
,
![y_n^2 \not= 1](/media/m/1/5/3/1535b54dd613f27be1a078d8f4c0a314.png)
,
![z_n^2 \not= 1](/media/m/f/1/e/f1e1772d8ad5f3ea94a92984d2584260.png)
.
![b)](/media/m/d/2/f/d2f292cd6a69e9158afe71ba9d830da4.png)
Postoji li
![k \in \mathbb{N}](/media/m/3/7/4/3740df5f7c5224aee0c08404b0d63c46.png)
takav da je
![x_k + y_k + z_k = 0](/media/m/b/d/8/bd852908a54ebf5f644a4bd6adb97d61.png)
?
%V0
Nizovi realnih brojeva $(x_n), (y_n), (z_n), n \in \mathbb{N}$, definirani su formulama $$x_{n+1}=\frac{2x_n}{x_n^2 - 1}\text{,}\,\,\, y_{n+1}=\frac{2y_n}{y_n^2 - 1}\text{,}\,\,\, z_{n+1}=\frac{2z_n}{z_n^2 - 1}\text{,}$$ a početni članovi su $x_1 = 2$, $y_1 = 4$ i $z_1$ takav da vrijedi $x_1y_1z_1 = x_1 + y_1 + z_1$.
$a)$ Provjerite da su za svaki $n \in \mathbb{N}$ zadovoljeni uvjeti: $x_n^2 \not= 1$, $y_n^2 \not= 1$, $z_n^2 \not= 1$.
$b)$ Postoji li $k \in \mathbb{N}$ takav da je $x_k + y_k + z_k = 0$?
Izvor: Državno natjecanje iz matematike 2004