2007 Cyprus MO/Lyceum/Problem 9

Problem

We consider the sequence of real numbers $\displaystyle a_1,a_2,a_3\ldots$ , such that $a_1=0$ , $a_2=1$ and $\displaystyle a_n=a_{n-1}-a_{n-2}$ , $\forall n \in \{3,4,5,6\ldots\}$ . The value of the term $\displaystyle a_{138}$ 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, and $\displaystyle 138\equiv0\pmod6$ . $a_{138}=-1\Longrightarrow\mathrm{ B}$

2007 Cyprus MO, Lyceum (Problems)
Preceded by Problem 8	Followed by Problem 10
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2007 Cyprus MO/Lyceum/Problem 9

Problem

Solution

See also