(adsbygoogle = window.adsbygoogle || []).push({}); [SOLVED] Cyclic Sequence of Angles

Fix an angle [itex]\theta[/itex]. Let n be a positive integer and define [itex]\theta_n = n\theta \bmod 2\pi[/itex].

The sequence [itex]\theta_1, \theta_2, \ldots[/itex] is cyclic if if it starts repeating itself at some point, i.e. the sequence has the form [itex]\theta_1, \ldots, \theta_k, \theta_1 \ldots[/itex].

What I would like to find out is: For which angles [itex]\theta[/itex] is the sequence [itex]\{\theta_n\}[/itex] cyclic? If for some integer m > 1, [itex]\theta_1 = \theta_m \equiv \theta = m\theta[/itex], then [itex]m\theta = \theta + 2\pi x[/itex] for some non-negative integer x. Solving for [itex]\theta[/itex], I get [itex]\theta = 2\pi x / (m - 1)[/itex]. So it seems that any rational multiple of [itex]\pi[/itex] will create a cyclic sequence. Is this correct?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Cyclic Sequence of Angles

**Physics Forums | Science Articles, Homework Help, Discussion**