2018/19 AHSMC Part 1 Problem 4

Problem 4 of 16

A sequence is defined by $a_0 = 0$ and $a_{n+1} = \dfrac{a_n - 3}{a_n - 2}$ for all integers $n \ge 0$ . Then $a_{2018}$ is

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We have $a_0 = 0$ , $a_1 = \dfrac{3}{2}$ , $a_2 = \dfrac{\frac{3}{2} - 3}{\frac{3}{2} - 2} = 3$ , $a_3 = 0$ , $\cdots$ . Hence the sequence repeats after $3$ steps and $a_{3k} = 0$ for all positive integers $k$ . Thus $a_{2016} = 0$ and hence $a_{2018} = 3$ . The answer is (D).

University of Alberta

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