# 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 $$999$$.)

×