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2007 Cyprus MO/Lyceum/Problem 9

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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}

See also

2007 Cyprus MO, Lyceum (Problems)
Preceded by
Problem 8 		Followed by
Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
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