Understanding Geometric Series: Concepts, Applications, and Formulas
This chapter delves into geometric series, defining them as the sums of geometric sequences. Key formulas and examples elucidate how to calculate the sum of various series, including the series 5 + 15 + 45 + ... and others. Practical applications are highlighted, such as analyzing bouncing balls and telephone fan-out scenarios. Students are guided through the process of using sigma notation for series and solving related problems. This comprehensive overview is essential for mastering geometric series and their real-world implications.
Understanding Geometric Series: Concepts, Applications, and Formulas
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Presentation Transcript
Chapter 6 Sequences and Series 6.5 Geometric Series 6.5.1 MATHPOWERTM 12, WESTERN EDITION
Geometric Series A geometric series is the sum of a geometric sequence. The formula for a geometric series is: Example: Find the sum of the series 5 + 15 + 45 + . . . + 10 935. 6.5.2
Geometric Series Find the sum of the first seven terms of the series 27 + 9 + 3 + . . .: 6.5.3
Geometric Series How many terms of the series 2 + (-4) + 8 + (-16) + . . . will yield a sum of 342? 6.5.4
Applications --The Bouncing Ball A ball is dropped from a height of 100 m and bounces back to 40% of its previous height. Find the height of the ball after it hits the floor for the fourth time. tn = arn - 1 The vertical height of the ball after the fourth bounce is. 6.5.5
The Bouncing Ball [cont’d] Find the total vertical distance travelled by the ball when it contacts the floor for the fifth time. The total vertical distance travelled is the sum of the upward and downward distances. The total vertical distance will be 6.5.6
Applications--The Telephone Fan-Out Level 1 20 Level 2 21 Level 3 22 • a) How many students will be contacted at the 8th level? • b) At what level will 64 students be contacted? • c) By the 8th, how many students will be • contacted altogether? • d) By the nth level, how many students will be • contacted altogether? • Suppose there are 300 students to be contacted. • By what level will all have been contacted? 6.5.7
The Telephone Fan-Out [cont’d] a) How many students will be contacted at the 8th level? b) At what level will 64 students be contacted? c) By the 8th level, how many students will be contacted altogether? d) By the nth level, how many students will be contacted altogether? e) Suppose there are 300 students to be contacted. By what level will all have been contacted? 6.5.8
Using Sigma Notation Write the following series using sigma notation and then find the sum of the series: 27 + 81 + 243 + 729 + 2187 + 6561 Summation notation for this series is: The sum of the series is 6.5.9
Assignment Suggested Questions: Pages 309 and 310 1-21 odd, 22, 23, 28, 32 a 6.5.10
