Balkan 90/1 solution

7th Balkan 1990

Problem 1

The sequence u_n is defined by u₁ = 1, u₂ = 3, u_n = (n+1) u_n-1 - n u_n-2. Which members of the sequence which are divisible by 11?

Solution

Answer: all n except 1, 2, 3, 5, 6, 7, 9.

We calculate the residues mod 11 to be: u₁ = 1, u₂ = 3, u₃ = -2, u₄ = 0, u₅ = -1, u₆ = 4, u₇ = 6, u₈ = 0, u₉ = 1, u₁₀ = 0, u₁₁ = 0. But now u_n = 0 mod 11 for all n ≥ 11. So u_n is divisible by 11 for n = 4, 8 and n ≥ 10.

7th Balkan 1990