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Chapter 6 Sequences and Series. 6.5. Geometric Series. 6.5. 1. MATHPOWER TM 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.

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## MATHPOWER TM 12, WESTERN EDITION

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**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. tn = arn - 1 10 935 = 5(3)n - 1 2187 = 3n - 1 37 = 3n - 1 7 = n - 1 8 = n Sn= 16 400 The sum of the series is 16 400. 6.5.2**Geometric Series**Find the sum of the first seven terms of the series 27 + 9 + 3 + . . .: The sum of the first seven terms is or approximately 40.5. 6.5.3**Geometric Series**How many terms of the series 2 + (-4) + 8 + (-16) + . . . will yield a sum of 342? -1026 = 2((-2)n - 1) - 513 = (-2)n - 1 -512 = (-2)n (-2)9 = (-2)n 9 = n For this geometric series, t9 = 342. Therefore, the sum of the first nine terms is 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 = 100(0.40)4 = 2.56 m The vertical height of the ball after the fourth bounce is 2.56 m. 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. 100 m The total vertical distance will be 2Sn - 100. Stotal = 2(164.96) - 100 = 229.92 The total vertical distance travelled is 229.92m. S5 = 164.96 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? e) 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? 28 - 1= 27or 128 students b) At what level will 64 students be contacted? 2n - 1 = 64 n = 7 at the 7th level c) By the 8th level, how many students will be contacted altogether? N.B. 255 = 254 students + 1 teacher 254 students S8 = 255 d) By the nth level, how many students will be contacted altogether? Sn = 2n - 1 e) Suppose there are 300 students to be contacted. By what level will all have been contacted? by the 9th level 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 tn = arn - 1 = 27(3)n - 1 = (33)(3n - 1) = 3n + 2 Summation notation for this series is: S6 = 9828 The sum of the series is 9828. 6.5.9**Assignment**Suggested Questions: Pages 309 and 310 1-21 odd, 22, 23, 28, 32 a 6.5.10

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