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GEOMETRIC SEQUENCES AND SERIES

GEOMETRIC SEQUENCES AND SERIES. GEOMETRIC SEQUENCES. A sequence is geometric if the ratios of consecutive terms are the same. “ r ” is called the common ratio of a geometric sequence. Some simple geometric sequences. 2, 4, 8, 16, ….. This sequence is geometric because

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GEOMETRIC SEQUENCES AND SERIES

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  1. GEOMETRIC SEQUENCES AND SERIES

  2. GEOMETRIC SEQUENCES • A sequence is geometric if the ratios of consecutive terms are the same. • “r” is called the common ratio of a geometric sequence.

  3. Some simple geometric sequences • 2, 4, 8, 16, ….. • This sequence is geometric because • 4/2 = 2 8/4 = 2 16/8 = 2, etc.. • In this sequence 2 is the common ratio (r = 2) • 12, 36, 108, 324, …. • 36/12 = 3 108/36 = 3 324/108 = 3 • r = 3

  4. The nth term • The nth term of a geometric sequence has the form

  5. Finding the terms of a geometric sequence • Write the first 4 terms of the geometric sequence with first term and common ratio given. • a1=6 and r = 4 • a1 = 6 a2 = 24 a3 = 96 a4 = 384 • a1=4 and r = -2 • a1= 4 a2 = -8 a3 = 16 a4 = -32

  6. Finding the nth term • Find a formula for the nth term and then find the indicated term. • 15th term; • Find r. • Write equation.

  7. GEOMETRIC SERIES

  8. Finite Geometric Series • The sum of a finite geometric sequence with common ratio r ≠ 1 is given by:

  9. Find the sum of each geometric series 1.

  10. 2.

  11. Infinite Geometric Series Think about it…. How is this even possible?

  12. Infinite Geometric Series

  13. Examples 1. No sum r > 1

  14. Geometric Sequence and Series Word Problems

  15. Example 1: Hot Tub Problem • The temperature in a hotel hot tub is increased by 5% each hour. If the present temperature of the hot tub is 85F, find the temperature of the hot tub after three hours. • If the temperature is increased by 5%, the new temperature will be 105% of the original temperature.

  16. The common ratio will be r = 1.05 • a1 = 85 • At the end of the 3rd hour or beginning of the 4th hour…

  17. Example 2: Exponential Growth • A culture of nasty virus doubles every 3 hours. If there are 300 virus at the beginning, how many virus will there be after 12 hours? • a1 = 300 r = 2

  18. Since the virus doubles every 3 hours, in the 12 hour time period it will have doubled 4 times from the initial amount. Therefore use n = 5.

  19. Example 3: Ball Drop • A tennis ball dropped from a height of 30 feet bounces 40% of the height from which it fell on each bounce. What is the vertical distance it travels before coming to rest? • Start the sequence at the first ball bounce: • a1 = 30 x .40 = 12

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