2YSequenceNotation

Sequence Notation

The solution to sin(x) = 0 can be written as a list: math . \qquad x = \{ \dots -\pi, \, 0, \, \pi, \, 2\pi, \, 3\pi, \, \dots \} math

or it can be written as a general solution using sequence notation math . \qquad x = n\pi, \quad n \in Z math

{recall that **Z** is the set of __integers__ (positive and negative). Z + would be the positive integers, not including zero}


 * NOTE: ** Some texts use **J** for the set of integers. **J** and **Z** are both acceptable.

Sequence notation is shorter and neater but it can sometimes be difficult to forumulate. The trick is to look for ways that increasing n by one can be transformed into the desired result.

Even numbers, such as the following math . \qquad x = \Big\{ \dfrac {2\pi}{3}, \; \dfrac {4\pi}{3}, \; \dfrac {6\pi}{3}, \; \dfrac {8\pi}{3}, \; \dots \Big\} math

can be represented using 2n ... math . \qquad x = \dfrac {2n\pi}{3}, \quad n \in Z^+ math

Odd numbers can be represented using 2n – 1. {Note that changing to a common denominator will often help} math . \qquad x = \Big\{ \dfrac {\pi}{6}, \; \dfrac {\pi}{2}, \; \dfrac {5\pi}{6}, \; \dfrac {7\pi}{6}, \; \dots \Big\} = \Big\{ \dfrac {\pi}{6}, \; \dfrac {3\pi}{6}, \; \dfrac {5\pi}{6}, \; \dfrac {7\pi}{6}, \; \dots \Big\} math becomes ... math . \qquad x = \dfrac {(2n-1)\pi}{6}, \quad n \in Z^+ math

The solution to sin(x)=±½ has a more complicated pattern math . \qquad x = \Big\{\dots \dfrac {-\pi}{6}, \; \dfrac {\pi}{6}, \; \dfrac {5\pi}{6}, \; \dfrac {7\pi}{6}, \; \dfrac {11\pi}{6}, \; \dfrac {13\pi}{6}, \; \dots \Big\} math

This can be transformed into sequence notation by considering how each value relates to the x-axis math . \qquad x = n\pi \pm \dfrac {\pi}{6}, \quad n \in Z math but writing solutions as two seperate patterns is perfectly acceptable.

Review exact values here. Review quadrants here.

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