A Little Hailstone

★ 8/10number theory Recursion sequences Induction

Problem

Fix a positive integer $m$ . A sequence of positive integers has $a_1 = 1$ and

a_{n+1} = \begin{cases} \dfrac{a_n}{2}, & a_n \text{ even}, \\[4pt] a_n + m, & a_n \text{ odd}. \end{cases}

1. For $m = 5$ , find $a_8, a_9, a_{10}$ . 2. Prove that $\{a_n\}$ is increasing iff $m$ is even. 3. For odd $m$ , does some $n > 1$ satisfy $a_n = 1$ ? Justify.

Answer

Solution

Difficulty	8/10
Topics	number theory, Recursion, sequences, Induction

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