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Sequences and fractional parts

Source: Romanian District Olympiad 2024 9.2

March 10, 2024
algebraSequencefractional part

Problem Statement

Consider the sequence (an)n⩾1(a_n)_{n\geqslant 1} defined by a1=1/2a_1=1/2 and 2n⋅an+1=(n+1)an.2n\cdot a_{n+1}=(n+1)a_n. [*]Determine the general formula for an.a_n. [*]Let bn=a1+a2+⋯+an.b_n=a_1+a_2+\cdots+a_n. Prove that {bn}−{bn+1}≠{bn+1}−{bn+2}.\{b_n\}-\{b_{n+1}\}\neq \{b_{n+1}\}-\{b_{n+2}\}.
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