2. Copyright © 2011 Pearson Education, Inc. Geometric Sequences and Series Section 8.3 Sequences, Series, and Probability

3. 11.3 Copyright © 2011 Pearson Education, Inc. Slide 11-3 A geometric sequence is defined as a sequence in which there is a constant ratio r between consecutive terms or by giving a general formula that will produce such a sequence. According to the definition, every geometric sequence has the following form: a, ar, ar 2, ar 3, ar 4,… Every term (after the first) is a constant multiple of the term preceding it. Of course, that constant multiple is the constant ratio r. Note that if a = r then a n = ar n –1 = rr n –1 = r n. Geometric Sequences

4. 11.3 Copyright © 2011 Pearson Education, Inc. Slide 11-4 Definition: Geometric Sequence A sequence with general term a n = ar n – 1 is called a geometric sequence with common ratio r, where r ≠ 1 and r ≠ 0. Geometric Sequences

5. 11.3 Copyright © 2011 Pearson Education, Inc. Slide 11-5 The indicated sum of the terms of a geometric sequence is called a geometric series. The “trick” to finding the sum of n terms of a geometric series is to change the signs and shift the terms so that most terms “cancel out” when the two equations are added. This method can also be used to find a general formula for the sum of a geometric series. Geometric Sequences

6. 11.3 Copyright © 2011 Pearson Education, Inc. Slide 11-6 Let S n represent the sum of the first n terms of the geometric sequence a n = ar n –1. Adding S n and –rS n eliminates most of the terms: So the sum of a geometric series can be found if we know the first term, the constant ratio, and the number of terms. Geometric Sequences

7. 11.3 Copyright © 2011 Pearson Education, Inc. Slide 11-7 Theorem: Sum of a finite Geometric Series If S n represents the sum of the first n terms of a geometric series with first term a and common ratio r (r ≠ 1), then Geometric Sequences

8. 11.3 Copyright © 2011 Pearson Education, Inc. Slide 11-8 In general, it can be proved that r n → 0 as n → ∞, provided that |r| < 1, and r n does not get close to 0 as n → ∞. So, if |r| < 1 and n is large, then S n is approximately a/(1 – r). Furthermore, by using more terms in the sum we can get a sum that is arbitrarily close to the number a/(1 – r). In this sense we say that the sum of all terms of the infinite geometric series is a/(1 – r). Infinite Geometric Sequences

9. 11.3 Copyright © 2011 Pearson Education, Inc. Slide 11-9 Theorem: Sum of an Infinite Geometric Series If a + ar + ar 2 + ··· is an infinite geometric series with |r| < 1, then the sum S of all of the terms is given by We can use the infinity symbol ∞ to indicate the sum of infinitely many terms of an infinite geometric series as follows: Infinite Geometric Sequences