Assignment Answers: Find the partial sum of the following: 1. = 250/2 ( ) = 218, = 101/2 (1/2 – 73/4) = Find the indicated n th partial sum of the following arithmetic sequence: 3. -6, -2, 2, 6, … n = 10 = 10/2 (–6 + 30) = , 29, 18, 7, … n = 12 = 12/2 (40 – 81) = – 246

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Warm-up: Find the 9th term of the geometric sequence 7, 21, 63,... Example: Finding the nth Term a 1 = 7 The 9th term is 45,927. a n = a 1 r n – 1 a 9 = 7(3) 9 – 1 = 7(3) 8 = 45,927

Geometric Series Unit 2 Concept 2 the sum of a geometric sequence

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 The sum of the first n terms of a sequence is represented by summation notation. Definition of Summation Notation index of summation upper limit of summation lower limit of summation

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 The Sum of a Finite Geometric Sequence The sum of a finite geometric sequence is given by = ? n = 8 a 1 = 5

INFINITE GEOMETRIC SERIES 6

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Definition of Geometric Series The sum of the terms of an infinite geometric sequence is called a geometric series. a 1 + a 1 r + a 1 r 2 + a 1 r a 1 r n If |r| < 1, then the infinite geometric series has the sum

Convergence vs Divergence Convergence – a series with a sum, -1 < r < 1 Divergence – a series without a sum, the sum approaches positive or negative infinity, r 1 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8

Do the following series converge or diverge? 1.a 1 = 8, r = … diverge, since r > 1 2.Converge, since r < 1 3.Converge, since r < 1

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Example: Sum of Infinite Geometric Series Example: Find the sum of The sum of the series is Sum or no Sum?

Assignment: Copyright © by Houghton Mifflin Company, Inc. All rights reserved /2 + 27/8 +… /3 - 1/9 +…