Question Of The Day

The sequence u_n is defined by u₁ = 1, u₂ = 3, u_n = (n + 1) u_{n – 1} – n u_{n – 2}. How many members of the sequence are not divisible by 11?  

a) 7

b) 8

c) 9

d) Infinite

Ans: a)  7

For n = 1, 2, 3, 5, 6, 7, 9 (Explanation; 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 & n ≥ 10.)