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Geometric Sequences and Series

Geometric Sequences and Series. Sections 11.3 and 11.5. Review Terms. 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

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Geometric Sequences and Series

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  1. Geometric Sequences and Series Sections 11.3 and 11.5

  2. Review Terms • 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

  3. 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

  4. 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.

  5. 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.

  6. 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

  7. Finite Series Practice Evaluate the following series for the given number of terms: • 1 + 2 + 4 + …; S8 S8= (1 (1 – 28))/(1 – 2) = 255

  8. Finite Series Practice Evaluate the following series for the given number of terms: • 1 + 2 + 4 + …; S8 S8= (1 (1 – 28))/(1 – 2) = 255 S5= (7 (1 – (– 5)5))/(1 – (–5)) = 3647

  9. 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

  10. Infinite Series Practice Evaluate the following geometric series, or find the fraction equivalent for the given infinite repeating decimal. 1.22222… • 0.222222… a1 = 0.2, r = 0.1 S = 2/9

  11. Practice Convert the following infinite repeating decimals to fractions. • 0.42857142857142… • 0.066666666… • 0.2727272727… 3/7 1/15 3/11

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