Problem 4 : 2 sequences

Algebra Level 5

Let \(n\) be a positive integer and let \(a_1, a_2, ..., a_{n - 1}\) be arbitary real numbers. Define the sequences \(u_0, u_1, u_2, ..., u_n\) and \(v_0, v_1, v_2, ..., v_n\) inductively by \(u_0 = u_1 = v_0 = v_1 = 1\) and \(u_{k + 1} = u_k + a_ku_{k - 1}, v_{k + 1} = v_k + a_{n - k}v_{k - 1}\), for \(k = 1, 2, ...., n - 1\). Determine the largest possible value of \(|u_n - v_n|\).

(If you think the answer is infinity, answer \(999\).)

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