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A sequence

Source: Irish Math Olympiad 2014

January 9, 2017
algebraSequence

Problem Statement

We say that a doubly infinite sequence ...,s−2,s−1,s0,s1,s2,.... . . , s_{−2}, s_{−1}, s_{0}, s_1, s_2, . . . is subaveraging if sn=(sn−1+sn+1)/4s_n = (s_{n−1} + s_{n+1})/4 for all integers n. (a) Find a subaveraging sequence in which all entries are different from each other. Prove that all entries are indeed distinct. (b) Show that if (sn)(s_n) is a subaveraging sequence such that there exist distinct integers m, n such that sm=sns_m = s_n, then there are infinitely many pairs of distinct integers i, j with si=sjs_i = s_j .
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