Question 134
$\{a_n\}_{n \ge 0}$ is an infinite arithmetic sequence of integers with common difference $d$, whose first term satisfies $1 \le a_0 \le d$. Call this sequence $S_0$; we define a sequence of new sequences recursively: the sequence $S_{n+1}$ is obtained from $S_n$ by the following two steps:
Step 1. Let $b_n$ be the first term of $S_n$. Remove the first term and shift every remaining term of the sequence forward.
Step 2. Starting from the new first term, add 1 to each of the first $b_n$ terms of the sequence.
Prove that there is a constant $c$ such that $b_n = [ca_n]$ for all $n \ge 0$. Here $[\cdot]$ denotes the floor function.
Step 1. Let $b_n$ be the first term of $S_n$. Remove the first term and shift every remaining term of the sequence forward.
Step 2. Starting from the new first term, add 1 to each of the first $b_n$ terms of the sequence.
Prove that there is a constant $c$ such that $b_n = [ca_n]$ for all $n \ge 0$. Here $[\cdot]$ denotes the floor function.