2007 Cyprus MO/Lyceum/Problem 9

Problem

We consider the sequence of real numbers $a_1,a_2,a_3\ldots$ , such that $a_1=0$ , $a_2=1$ and $a_n=a_{n-1}-a_{n-2}$ , $\forall n \in \{3,4,5,6\ldots\}$ . The value of the term $a_{138}$ is

$\mathrm{(A) \ } 0\qquad \mathrm{(B) \ } -1\qquad \mathrm{(C) \ } 1\qquad \mathrm{(D) \ } 2\qquad \mathrm{(E) \ } -2$

Solution

The first few terms of the sequence are:

0, 1, 1, 0, -1, -1, 0, 1, 1, 0 ...

The sequence repeats every 6 terms.

$138\equiv0\mod6$

$a_{138}=-1\Rightarrow\mathrm{ B}$

Page

Toolbox

Search

2007 Cyprus MO/Lyceum/Problem 9

Problem

Solution

See also