IM Alg2.1.5 Lesson: Sequences are Functions

Describe how to produce one step of the pattern from the previous step.
[img]data:image/png;base64,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[/img]
Here is a visual pattern of dots.
[size=150]The number of dots[math]D(n)[/math]  is a function of the step number [math]n[/math].[/size][br][br][img]data:image/png;base64,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[/img][br]What values make sense for [math]n[/math] in this situation?
What values don't make sense for [math]n[/math]?
Complete the table for Steps 1 to 5.
Following the pattern in the table, write an equation for [math]D(n)[/math] in terms of the previous step. Be prepared to explain your reasoning.[br]
Consider the same triangular pattern.
Is the sequence defined by the number of dots in each step arithmetic, geometric, or neither? Explain how you know.
Can you write an expression for the number of dots in Step [math]n[/math] without using the value of [math]D[/math] from a previous step?
Use the first 5 terms of each sequence to state if the sequence is arithmetic, geometric, or neither.
[size=150]Next, define the sequence recursively using function notation.[/size][br][br][math]30,40,50,60,70,...[/math]
[math]80,40,20,10,5,2.5,...[/math]
[math]1,2,4,8,16,32,...[/math]
[math]1,\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16},...[/math]
[math]20,13,6,\text{-}1,\text{-}8,...[/math]
[math]1,3,7,15,31,...[/math]
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Information: IM Alg2.1.5 Lesson: Sequences are Functions