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a_{n+1}=(a_n+b] (VI Soros Olympiad 1990-00 R2 9.5)

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May 28, 2024
number thoeryalgebraSequence

Problem Statement

Let b be a given real number. The sequence of integers a1,a2,a3,...a_1, a_2,a_3, ... is such that a1=(b]a_1 =(b] and an+1=(an+b]a_{n+1}=(a_n+b] for all n≥1n\ge 1 Prove that the sum a1+a22+a33+...+anna_1+\frac{a_2}{2}+\frac{a_3}{3}+...+\frac{a_n}{n} is an integer number for any natural nn .
(In the condition of the problem, (x](x] denotes the smallest integer that is greater than or equal to xx)
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