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x_n = (p + \sqrt{q})^n - [(p + \sqrt{q})^n]

Source: Polish MO Second Round 1973 p4

September 8, 2024
algebralimit

Problem Statement

Let xn=(p+q)n−[(p+q)n] x_n = (p + \sqrt{q})^n - [(p + \sqrt{q})^n] for n=1,2,3,… n = 1, 2, 3, \ldots . Prove that if p p , q q are natural numbers satisfying the condition p−1<q<p p - 1 < \sqrt{q} < p , then lim⁡n→∞xn=1 \lim_{n\to \infty} x_n = 1 .
Attention. The symbol [a] [a] denotes the largest integer not greater than a a .
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