MathDB
Interesting recurring sequence with floor function

Source: 2017 Taiwan TST Round 3

April 13, 2018
arithmetic sequencealgebrafloor functionfunction

Problem Statement

Let {an}n≥0\{a_n\}_{n\geq 0} be an arithmetic sequence with difference dd and 1≤a0≤d1\leq a_0\leq d. Denote the sequence as S0S_0, and define SnS_n recursively by two operations below: Step 11: Denote the first number of SnS_n as bnb_n, and remove bnb_n. Step 22: Add 11 to the first bnb_n numbers to get Sn+1S_{n+1}. Prove that there exists a constant cc such that bn=[can]b_n=[ca_n] for all n≥0n\geq 0, where [][] is the floor function.
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