MathDB
Sequence with floor function

Source: Romanian District Olympiad 2024 11.2

March 10, 2024
Sequenceslimitanalysis

Problem Statement

Let k⩾2k\geqslant 2 be an integer. Consider the sequence (xn)n⩾1(x_n)_{n\geqslant 1} defined by x1=a>0x_1=a>0 and xn+1=xn+⌊k/xn⌋x_{n+1}=x_n+\lfloor k/x_n\rfloor for n⩾1.n\geqslant 1. Prove that the sequence is convergent and determine its limit.
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