Areteem: MCIV (Chapt 9 Example 13)

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MCIV (Chapt 9 Example 13)

Hello,

After solving example 13, I come to realize that a lot of the periodic sequences can be represented by some sort of trigonometric function, i.e. when the solutions of the characteristic equation is complex. Is it true that whenever the solution of the characteristic equation is complex, then the sequence is periodic? Moreover, if the solution is the kth root of unity, can we say that the sequence is periodic by every k terms?

Thank you,

Tina Jin

If the roots are not on the unit circle, then the sequence cannot be periodic. For the case where the roots are the roots of unity, you can answer the question by thinking about how the recurrence relation will look like in that situation.

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Dear Dr. Wang,

Thank you for your reply! Why is it that if the roots are not on the unit circle, then the sequence is not periodic? It will still "wrap around" and repeat right?

Thank you,

Tina Jin

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If the root is not on the unit circle, say it is $r(\cos\theta+ i\sin\theta)$ where $r\neq 1$, then the sequence is $r^n(\cos(n\theta)+i\sin(n\theta))$. The modulus $r^n$ is either increasing or decreasing, so it cannot repeat.

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Hello Dr. Kevin Wang,

Thanks! I got it!

Tina Jin

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