1. Assignment Answers: Find the partial sum of the following: 1. = 250/2 (999 + 750) = 218,625 2. = 101/2 (1/2 – 73/4) = - 896.375 Find the indicated n th partial sum of the following arithmetic sequence: 3. -6, -2, 2, 6, … n = 10 = 10/2 (–6 + 30) = 120 4. 40, 29, 18, 7, … n = 12 = 12/2 (40 – 81) = – 246

2. 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

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

4. 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

5. 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 5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ? n = 8 a 1 = 5

6. INFINITE GEOMETRIC SERIES 6

7. 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 3 +... + a 1 r n-1 +... If |r| < 1, then the infinite geometric series has the sum

8. 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

9. Do the following series converge or diverge? 1.a 1 = 8, r = 5 2.64 + 16 + 4 + 1 + … 3. 9 1.diverge, since r > 1 2.Converge, since r < 1 3.Converge, since r < 1

10. 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?

11. Assignment: 1. 2. 3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 4. 5.8 + 6 + 9/2 + 27/8 +… 6.3 -1 + 1/3 - 1/9 +…