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find limit of a sequence with recurrence relation

Source: 2001 Moldova MO Grade 11 P2

April 13, 2021
recurrence relationSequencealgebra

Problem Statement

Let m≥2m\ge2 be an integer. The sequence (an)n∈N(a_n)_{n\in\mathbb N} is defined by a0=0a_0=0 and an=⌊nm⌋+a⌊nm⌋a_n=\left\lfloor\frac nm\right\rfloor+a_{\left\lfloor\frac nm\right\rfloor} for all nn. Determine lim⁡n→∞ann\lim_{n\to\infty}\frac{a_n}n.
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