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Found problems: 1

2010 Gheorghe Vranceanu, 4
Tags: number theory , algebra , seuqences , Periodicity
Let be two real numbers $ \alpha ,\beta $ and two sequences $ \left(x_n \right)_{n\ge 1} ,\left(y_n \right)_{n\ge 1} $ whose smallest periods are $ p,q, $ respectively. Prove that the sequence $ \left( \alpha x_n+\beta y_n\right)_{n\ge 1} $ is periodic if $ \text{gcd}^2 (p,q) | \text{lcm} (p,q) , $ and in this case find its smallest period.