Mala olimpijada 1998 zadatak 4
Dodao/la:
mljulj12. travnja 2012. Niz
![\{ a_n \}](/media/m/e/6/3/e6360b662cc481390fdd0e22cb947840.png)
je zadan na ovaj način:
![a_0=0, \ a_1=1, \ a_n=2a_{n-1}+a_{n-2}, \ n>1.](/media/m/e/6/5/e65c460327daf836f0219448ea3bf249.png)
Dokažite da
![2^k](/media/m/e/f/a/efa8b263b195099069a7f7883dd4938d.png)
dijeli
![a_n](/media/m/1/f/f/1ff6f81c68b9c6fb726845c9ce762d7a.png)
ako i samo ako
![2^k](/media/m/e/f/a/efa8b263b195099069a7f7883dd4938d.png)
dijeli
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
.
%V0
Niz $\{ a_n \} $ je zadan na ovaj način:
$$ a_0=0, \ a_1=1, \ a_n=2a_{n-1}+a_{n-2}, \ n>1.$$
Dokažite da $2^k$ dijeli $a_n$ ako i samo ako $2^k$ dijeli $n$.
Izvor: Mala olimpijada 1998 zadatak 2