Geometric Series.

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Presentation transcript:

Geometric Series

Use the formula to evaluate the series 5 + 15 + 45 + 135 + 405 + 1215. Geometric Series Lesson 11-5 Additional Examples Use the formula to evaluate the series 5 + 15 + 45 + 135 + 405 + 1215. The first term is 5, and there are six terms in the series. The common ratio is = = = = = 3 15 5 45 135 405 1215 So a1 = 5, r = 3, and n = 6. Sn = Write the formula. a1 (1 – r n) 1 – r = Substitute a1 = 5, r = 3, and n = 6. 5 (1 – 36) 1 – 3 = = 1820 Simplify. –3640 –2 The sum of the series is 1820.

Write the formula for the sum of a geometric series. Lesson 11-5 Additional Examples The Floyd family starts saving for a vacation that is one year away. They start with $125. Each month they save 8% more than the previous month. How much money will they have saved 12 months later? Relate: Sn = a1 (1 – r n) 1 – r Write the formula for the sum of a geometric series. Define: Sn = total amount saved a1 = 125 Initial amount. r = 1.08 Common ratio. n = 12 Number of months. Write: S12 = 125 ( 1 – 1.08 12) 1 – 1.08 2372.14 Use a calculator. Substitute. The amount of money the Floyd’s will have saved will be $2372.14.

Geometric Series Lesson 11-5 Additional Examples Decide whether each infinite geometric series diverges or converges. Then determine whether the series has a sum.   n = 1 2 3 n a. b. 2 + 6 + 18 + . . . a1 = = , a2 = = 2 3 1 4 9 a1 = 2, a2 = 6 r = ÷ = 4 9 2 3 r = 6 ÷ 2 = 3 Since | r | < 1, the series converges, and the series has a sum. Since | r | 1, the series diverges, and the series does not have a sum. >

The total distance that the pendulum swings through is 200 in. Geometric Series Lesson 11-5 Additional Examples The weight at the end of a pendulum swings through an arc of 30 inches on its first swing. After that, each successive swing is 85% of the length of the previous swing. What is the total distance the weight will swing by the time it comes to rest? The largest arc the pendulum swings through is on the first swing of 30 in., so a1 = 30. S = Use the formula. a1 1 – r = Substitute. 30 1 – 0.85 = 200 Simplify. The total distance that the pendulum swings through is 200 in.