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2006 Cyprus MO/Lyceum/Problem 20

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Problem

The sequence f:N \to R satisfies f(n)=f(n-1)-f(n-2),\forall n\geq 3. Given that f(1)=f(2)=1, then f(3n) equals

\mathrm{(A)}\ 3\qquad\mathrm{(B)}\ -3\qquad\mathrm{(C)}\ 2\qquad\mathrm{(D)}\ 1\qquad\mathrm{(E)}\ 0

Solution

Lets write out a couple of terms: \begin{align*}f(3)&=f(2)-f(1)=0&f(4)&=f(3)-f(2)=-1\\f(5)&=f(4)-f(3)=-1&f(6)&=f(5)-f(4)=0\end{align*} We quickly see that every third term is zero, so the answer is \mathrm{E}.

See also

2006 Cyprus MO, Lyceum (Problems)
Preceded by
Problem 19 		Followed by
Problem 21
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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