 Presentation on theme: "Copyright © 2007 Pearson Education, Inc. Slide 8-1."— Presentation transcript:

Copyright © 2007 Pearson Education, Inc. Slide 8-2 Chapter 8: Further Topics in Algebra 8.1Sequences and Series 8.2Arithmetic Sequences and Series 8.3Geometric Sequences and Series 8.4The Binomial Theorem 8.5Mathematical Induction 8.6Counting Theory 8.7Probability

Copyright © 2007 Pearson Education, Inc. Slide 8-3 1, 2, 4, 8, 16 … is an example of a geometric sequence with first term 1 and each subsequent term is 2 times the term preceding it. The multiplier from each term to the next is called the common ratio and is usually denoted by r. 8.3 Geometric Sequences and Series A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a constant nonzero real number.

Copyright © 2007 Pearson Education, Inc. Slide 8-4 8.3 Finding the Common Ratio In a geometric sequence, the common ratio can be found by dividing any term by the term preceding it. The geometric sequence 2, 8, 32, 128, … has common ratio r = 4 since

Copyright © 2007 Pearson Education, Inc. Slide 8-5 8.3 Geometric Sequences and Series nth Term of a Geometric Sequence In the geometric sequence with first term a 1 and common ratio r, the nth term a n, is

Copyright © 2007 Pearson Education, Inc. Slide 8-6 8.3 Using the Formula for the nth Term Example Find a 5 and a n for the geometric sequence 4, –12, 36, –108, … Solution Here a 1 = 4 and r = 36/ –12 = – 3. Using n=5 in the formula In general

Copyright © 2007 Pearson Education, Inc. Slide 8-7 8.3 Modeling a Population of Fruit Flies Example A population of fruit flies grows in such a way that each generation is 1.5 times the previous generation. There were 100 insects in the first generation. How many are in the fourth generation. Solution The populations form a geometric sequence with a 1 = 100 and r = 1.5. Using n=4 in the formula for a n gives or about 338 insects in the fourth generation.

Copyright © 2007 Pearson Education, Inc. Slide 8-8 8.3 Geometric Series A geometric series is the sum of the terms of a geometric sequence. In the fruit fly population model with a 1 = 100 and r = 1.5, the total population after four generations is a geometric series:

Copyright © 2007 Pearson Education, Inc. Slide 8-9 8.3 Geometric Sequences and Series Sum of the First n Terms of an Geometric Sequence If a geometric sequence has first term a 1 and common ratio r, then the sum of the first n terms is given by where.

Copyright © 2007 Pearson Education, Inc. Slide 8-10 8.3 Finding the Sum of the First n Terms Example Find Solution This is the sum of the first six terms of a geometric series with and r = 3. From the formula for S n,.

Copyright © 2007 Pearson Education, Inc. Slide 8-11 8.3 Infinite Geometric Series If a 1, a 2, a 3, … is a geometric sequence and the sequence of sums S 1, S 2, S 3, …is a convergent sequence, converging to a number S . Then S  is said to be the sum of the infinite geometric series

Copyright © 2007 Pearson Education, Inc. Slide 8-12 8.3 An Infinite Geometric Series Given the infinite geometric sequence the sequence of sums is S 1 = 2, S 2 = 3, S 3 = 3.5, … The calculator screen shows more sums, approaching a value of 4. So

Copyright © 2007 Pearson Education, Inc. Slide 8-13 8.3 Infinite Geometric Series Sum of the Terms of an Infinite Geometric Sequence The sum of the terms of an infinite geometric sequence with first term a 1 and common ratio r, where –1 < r < 1 is given by.

Copyright © 2007 Pearson Education, Inc. Slide 8-14 8.3 Finding Sums of the Terms of Infinite Geometric Sequences Example Find Solution Here and so.