2-5 Infinite Geometric SeriesRecursion & Special Sequences

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1Definitions & Equations

Writing & Solving Geometric Series

Practice Problems

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Definitions

Infinite Geometric Series A geometric series that has no final value

Recursive Formula Each term is formulated from one or more of the previous terms

Fibonacci Sequence Each term is formulated by adding the two previous terms 1, 1, 2, 3, 5, 8, 13, 21, 34… an = an-2 + an-1

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Sum of an Infinite Geometric Series

The sum (S) of an infinite geometric series with -1 < r < 1 is given by

If |r|≥1, the sum does not exist

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1

aS

r

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Finding the Sum of an Infinite Geometric Series

Find the sum if it exits

1 3 9...

2 8 32

1

1

2a

3 381 4

2

r

31, therefore the sum exists

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1

1

aS

r

1 12 2

3 11

4 4

S

2S

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Finding the Sum of an Infinite Geometric Series

Find the sum if it exits

1 2 4 8 ...

1 1a 22

1r

2 1, therefore the does notsum exists

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Use a Recursive Formula

Find the first five terms of a sequence in which a1=4 and an+1=3an-2, n≥1.

1 3 2n na a

11 13 2a a

2 3(4) 2 10a

12 23 2a a

3 3(10) 2 28a

13 33 2a a

4 3(28) 2 82a

14 43 2a a

5 3(82) 2 244a

4,10,28,82,244

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Fibonacci Sequences in Nature

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More Fibonacci Sequences in Nature

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My Personal Favorite Fibonacci Sequence in Nature

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Practice Problems

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Practice Problems