Problem 4 : 2 sequences

Algebra Level 5

Let nn be a positive integer and let a1,a2,...,an1a_1, a_2, ..., a_{n - 1} be arbitary real numbers. Define the sequences u0,u1,u2,...,unu_0, u_1, u_2, ..., u_n and v0,v1,v2,...,vnv_0, v_1, v_2, ..., v_n inductively by u0=u1=v0=v1=1u_0 = u_1 = v_0 = v_1 = 1 and uk+1=uk+akuk1,vk+1=vk+ankvk1u_{k + 1} = u_k + a_ku_{k - 1}, v_{k + 1} = v_k + a_{n - k}v_{k - 1}, for k=1,2,....,n1k = 1, 2, ...., n - 1. Determine the largest possible value of unvn|u_n - v_n|.

(If you think the answer is infinity, answer 999999.)

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