MathDB
Sequence of Points

Source: AMC 12 2008A #25

February 17, 2008
analytic geometrygeometrygeometric transformationdilationrotationtrigonometrygeometric sequence

Problem Statement

A sequence (a1,b1) (a_1,b_1), (a2,b2) (a_2,b_2), (a3,b3) (a_3,b_3), … \ldots of points in the coordinate plane satisfies (a_{n \plus{} 1}, b_{n \plus{} 1}) \equal{} (\sqrt {3}a_n \minus{} b_n, \sqrt {3}b_n \plus{} a_n)\hspace{3ex}\text{for}\hspace{3ex} n \equal{} 1,2,3,\ldots. Suppose that (a_{100},b_{100}) \equal{} (2,4). What is a_1 \plus{} b_1?
<spanclass=′latex−bold′>(A)</span>minus1297<spanclass=′latex−bold′>(B)</span>minus1299<spanclass=′latex−bold′>(C)</span> 0<spanclass=′latex−bold′>(D)</span> 1298<spanclass=′latex−bold′>(E)</span> 1296 <span class='latex-bold'>(A)</span>\\minus{} \frac {1}{2^{97}} \qquad <span class='latex-bold'>(B)</span>\\minus{} \frac {1}{2^{99}} \qquad <span class='latex-bold'>(C)</span>\ 0 \qquad <span class='latex-bold'>(D)</span>\ \frac {1}{2^{98}} \qquad <span class='latex-bold'>(E)</span>\ \frac {1}{2^{96}}
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