# Cyclic Sequence of Angles

• e(ho0n3
In summary, the sequence \theta_1, \ldots, \theta_k, \theta_1 \ldots is cyclic if if it starts repeating itself at some point, i.e. the sequence has the form \theta_1, \ldots, \theta_k, \theta_1 \ldots.

#### e(ho0n3

[SOLVED] Cyclic Sequence of Angles

Fix an angle $\theta$. Let n be a positive integer and define $\theta_n = n\theta \bmod 2\pi$.

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

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

Yes. Any rational multiple of pi creates a cyclic sequence. Why are you insecure about this?

This all began when I started contemplating about the limit as n approaches infinity of zn, z being complex with |z| < 1. If I represent z as r(cos t + isin t), zn = rn(cos nt + isin nt). If {nt} is cyclic, then I could break up the sine and cosine terms, multiply through by rn and apply the limit on each term. Each term goes to 0 because r < 1 so the limit is 0. Right?

Yes, but you don't have to worry about 'cyclic' |cos nt+i*sin nt| is bounded, since |cos|<=1 and |sin|<=1. Regardless of the arguments. So if you multiply by r^n with r<1, the result certainly goes to 0.

That makes sense. So for any complex z with |z| < 1, zn goes to 0 as n goes to infinity. I began feeling paranoid about this when I was trying to compute the limit of nzn as n goes to infinity. I rewrote this as n/z-n and applied l'Hopital's rule to get -zn/log z. I wasn't sure about the limit of zn here, but now I am. Thanks.

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## What is a cyclic sequence of angles?

A cyclic sequence of angles is a set of angles that are measured in a circular pattern, where the starting and ending points are the same. This creates a complete cycle or rotation.

## What are the properties of a cyclic sequence of angles?

The properties of a cyclic sequence of angles include: all angles add up to 360 degrees, the opposite angles are supplementary, and the sum of any two adjacent angles is equal to the third angle.

## What is the formula for finding the measure of an angle in a cyclic sequence?

The formula for finding the measure of an angle in a cyclic sequence is: measure of angle = (360/number of angles) * number of angles between the given angle and the starting point.

## How can a cyclic sequence of angles be used in real-life applications?

Cyclic sequences of angles are commonly used in navigation, astronomy, and engineering. They can also be used in designing circular objects, such as wheels or gears.

## What is the difference between a cyclic sequence of angles and a regular polygon?

A cyclic sequence of angles is a continuous set of angles that form a complete rotation, while a regular polygon is a closed shape with equal sides and angles. A cyclic sequence can have any number of angles, while a regular polygon typically has a fixed number of angles.

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