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11.4 Geometric Sequences

11.4 Geometric Sequences. 11.4 Geometric Sequences and Series. If we start with a number, a 1 , and repeatedly multiply it by some constant, r , then we have a geometric sequence : a 1 , a 1 r , a 1 r 2 , a 1 r 3 , a 1 r 4 ,…. The n th term of a geometric sequence is given by

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11.4 Geometric Sequences

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  1. 11.4 Geometric Sequences

  2. 11.4 Geometric Sequences and Series • If we start with a number, a1, and repeatedly multiply it by some constant, r, then we have a geometric sequence: a1, a1r, a1r2, a1r3, a1r4,…. • The nth term of a geometric sequence is given by • The number r is called the common ratio

  3. 11.4 Geometric Sequences and Series (Example 2) Find the 8th term of the geometric sequence: 5, 15, 45, … Solution: Use formula, an = a1r(n – 1) a1 = r = n = (Now work Example 3 in text….)

  4. Geometric Series • The sum of the terms of a geometric sequence is called a geometric series. • For example: is a finite geometric series with common ratio r. • What is the sum of the first n terms of a finite geometric series?

  5. Deriving a formula for the nth partial sum of a geometric sequence

  6. 11.4 Geometric Sequences and Series • The sum of the terms of an infinite geometric sequence is an infinite geometric series. • For some geometric sequences, Sn gets close to a specific number as n gets large. • This number becomes the limit of the sum of the infinite geometric sequence. • When |r|<1, the limit or sum of an infinite geometric series is given by .

  7. 11.4 Geometric Sequences and Series • You should be able to: • Identify the common ratio of a geometric sequence, and find a given term and the sum of the first n terms. • Find the sum of an infinite geometric series, if it exists.

  8. 11.4 Sequences and Series • You should be able to: • Find terms of sequences given the nth term. • Convert between sigma notation and other notation for a series.

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