2018/19 AHSMC Part 1 Problem 12

Problem 12 of 16

Let $a_n = n^2 + 2n + 50$ , $n = 1,2,\ldots$ . Let $d_n$ be the largest positive integer that is a divisor of both $a_n$ and $a_{n+1}$ . The maximum value of $d_n$ , $n = 1,2,\ldots$ is

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$d_n$ must also divide $a_{n+1} - a_n = 2n + 3$ , and since $4a_n = 4n^2 + 8n + 200 = (2n+3)^2 - 2(2n+3) + 197$ , $d_n$ must divide $197$ . Hence, the largest value of $d_n$ is $197$ , which is obtained for $n = 97$ . The answer is (C).

University of Alberta

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