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**Geometric Sequences and Series**

Sections 11.3 and 11.5

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**Review Terms Sequence Series Term Arithmetic Sequence**

An ordered list of numbers Series The sum of the terms of a sequence Term A specific number in a sequence Arithmetic Sequence A sequence of numbers where the difference between consecutive terms is constant Geometric Sequence A sequence of numbers where the ratio between consecutive terms is constant

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**Geometric Equations Recursive Closed (or explicit)**

This equation refers to other terms in the sequence. an = an–1 ∙ r This equation allows you to find any term in the sequence directly. an = a1 ∙r n–1

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Geometric sequence Determine if the following sequence is geometic. If it is, write both types of formulas. 1, –2, 4, –8, … Yes this is. an = an–1∙ (–2) and an = (–2)n–1 1, 2, 3, 4, … No this is not geometric. The ratios keep changing.

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Practice Determine if each of the following sequences is geometric. If it is write both types of formulas. 7, 0.7, 0.07, 0.007, … 10, 15, 22.5, 33.75, … 1/2, 1/4, 1/6, 1/8, … Yes. an = an–1•(0.1) or an = 7 (0.1)n–1 Yes. an = an–1 •(1.5) or an = 10 (1.5)n–1 No, there is not a common ratio.

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**Finite Geometric Series**

This is used for finite geometric series. n is the number of terms a1 is the first term in the sequence r is the common ratio between consecutive terms

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**Finite Series Practice**

Evaluate the following series for the given number of terms: …; S8 S8 = (1 (1 – 28))/(1 – 2) = 255

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**Finite Series Practice**

Evaluate the following series for the given number of terms: …; S8 S8 = (1 (1 – 28))/(1 – 2) = 255 S5 = (7 (1 – (– 5)5))/(1 – (–5)) = 3647

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**Infinite Geometric Series**

This is used for infinite geometric series The variables are the same as for the finite series This can be used to convert repeating decimals to fractions

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**Infinite Series Practice**

Evaluate the following geometric series, or find the fraction equivalent for the given infinite repeating decimal. … … a1 = 0.2, r = 0.1 S = 2/9

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Practice Convert the following infinite repeating decimals to fractions. … … … 3/7 1/15 3/11

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