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Paper foldingObserve what happens as I rip and generously share portions of my paper with you.
____ + _____ + ______ +______+_____+ ……. = ________
Is this an arithmetic series? Why or why not?
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Geometric SeriesIn a geometric series the terms form a _________________________.
The terms of a geometric sequence for a ________________________.
What is the common ratio?Example: 4+ 12 + 36+……..
The common ratio of the terms is _________.
Unlike an arithmetic series, a geometric series can approach a certain number as more terms are added to it.
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Converging vs. Diverging• Any geometric series that converges must have a common ratio less then
one.
• Any geometric series that diverges will have a common ratio greater then one.
To summarize:
Converges if: Diverges if:
|r| < 1Will approach a certain number.
|r| ≥ 1Will get infinitely big (or
infinitely small)
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Note: A diverging series that has r ≥ 1 diverges, however if |r| is close to one but still greater then 1, it can diverge really slowly.
for example r=1.01 in the series below:→ 1+1.01+1.0201+1.030301+…….
grows slowly but since every new term grows the series by more then one, the more terms that get added on the bigger the number will become.
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Do the following infinite geometric series diverge or converge?
1) 1+
Common ratio:____________________________
2)
Common ratio:_____________________________
3) 4+2+1+………?
Common ratio:_____________________________
Converge or diverge?
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Sum of a Geometric SeriesTo find the sum of a finite geometric series you can use the formula:
Where is the first number in the sequence, r is the common ration and n is the number of terms.
Example: Use the formula to evaluate the following series to 8 terms: 5+15+45+135+……….,n=8
Common ratio:_____________
𝑆𝑛=𝑎1(1−𝑟𝑛) 1−𝑟
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More Examples1) Calculate the number of triangles in a Sierpinski triangle
after stage 15.
2) Calculate the area of each additional triangle after stage 15