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Part of 1997 Baltic Way

Problems(1)

Sequence is eventually periodic

Source: Baltic Way 1997

1/28/2011
In a sequence u0,u1,…u_0,u_1,\ldots of positive integers, u0u_0 is arbitrary, and for any non-negative integer nn, un+1={12unfor even una+unfor odd un u_{n+1}=\begin{cases}\frac{1}{2}u_n & \text{for even }u_n \\ a+u_n & \text{for odd }u_n \end{cases} where aa is a fixed odd positive integer. Prove that the sequence is periodic from a certain step.
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