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Geometric

Geometric. Sequences & Series By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org. Last Updated: March 23, 2011. Geometric Sequences. 1, 2, 4, 8, 16, 32, … 2 n-1 , … 3, 9, 27, 81, 243, … 3 n , . . . 81, 54, 36, 24, 16, … ,. Jeff Bivin -- LZHS.

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Geometric

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  1. Geometric Sequences & Series By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: March 23, 2011

  2. Geometric Sequences 1, 2, 4, 8, 16, 32, … 2n-1, … 3, 9, 27, 81, 243, … 3n, . . . 81, 54, 36, 24, 16, … , . . . Jeff Bivin -- LZHS

  3. nth term of geometric sequence an = a1·r(n-1) Jeff Bivin -- LZHS

  4. Find the nth term of thegeometric sequence First term is 2 Common ratio is 3 an = a1·r(n-1) an = 2(3)(n-1) Jeff Bivin -- LZHS

  5. Find the nth term of a geometric sequence First term is 128 Common ratio is (1/2) an = a1·r(n-1) Jeff Bivin -- LZHS

  6. Find the nth term of the geometric sequence First term is 64 Common ratio is (3/2) an = a1·r(n-1) Jeff Bivin -- LZHS

  7. Finding the 10th term a1 = 3 r = 2 n = 10 3, 6, 12, 24, 48, . . . an = a1·r(n-1) a10 = 3·(2)10-1 a10 = 3·(2)9 a10 = 3·(512) a10 = 1536 Jeff Bivin -- LZHS

  8. Finding the 8th term a1 = 2 r = -5 n = 8 2, -10, 50, -250, 1250, . . . an = a1·r(n-1) a8 = 2·(-5)8-1 a8 = 2·(-5)7 a8 = 2·(-78125) a8 = -156250 Jeff Bivin -- LZHS

  9. Sum it up Jeff Bivin -- LZHS

  10. 1 + 3 + 9 + 27 + 81 + 243 a1 = 1 r = 3 n = 6 Jeff Bivin -- LZHS

  11. 4 - 8 + 16 - 32 + 64 – 128 + 256 a1 = 4 r = -2 n = 7 Jeff Bivin -- LZHS

  12. Alternative Sum Formula We know that: Multiply by r: Simplify: Substitute: Jeff Bivin -- LZHS

  13. Find the sum of the geometric Series Jeff Bivin -- LZHS

  14. Evaluate = 2 + 4 + 8+…+1024 a1 = 2 r = 2 n = 10 an = 1024 Jeff Bivin -- LZHS

  15. Evaluate = 3 + 6 + 12 +…+ 384 a1 = 3 r = 2 n = 8 an = 384 Jeff Bivin -- LZHS

  16. Review -- Geometric Sum of n terms nth term an = a1·r(n-1) Jeff Bivin -- LZHS

  17. Geometric Infinite Series Jeff Bivin -- LZHS

  18. The Magic Flea(magnified for easier viewing) There is no flea like a Magic Flea Jeff Bivin -- LZHS

  19. The Magic Flea(magnified for easier viewing) Jeff Bivin -- LZHS

  20. Sum it up -- Infinity Jeff Bivin -- LZHS

  21. Remember --The Magic Flea Jeff Bivin -- LZHS

  22. Jeff Bivin -- LZHS

  23. A Bouncing Ball rebounds ½ of the distance from which it fell -- What is the total vertical distance that the ball traveled before coming to rest if it fell from the top of a 128 feet tall building? 128 ft 64 ft 32 ft 16 ft 8 ft Jeff Bivin -- LZHS

  24. A Bouncing Ball Downward = 128 + 64 + 32 + 16 + 8 + … 128 ft 64 ft 32 ft 16 ft 8 ft Jeff Bivin -- LZHS

  25. A Bouncing Ball Upward = 64 + 32 + 16 + 8 + … 128 ft 64 ft 32 ft 16 ft 8 ft Jeff Bivin -- LZHS

  26. A Bouncing Ball Downward = 128 + 64 + 32 + 16 + 8 + … = 256 Upward = 64 + 32 + 16 + 8 + … = 128 TOTAL = 384 ft. 128 ft 64 ft 32 ft 16 ft 8 ft Jeff Bivin -- LZHS

  27. A Bouncing Ball rebounds 3/5 of the distance from which it fell -- What is the total vertical distance that the ball traveled before coming to rest if it fell from the top of a 625 feet tall building? 625 ft 375 ft 225 ft 135 ft 81 ft Jeff Bivin -- LZHS

  28. A Bouncing Ball Downward = 625 + 375 + 225 + 135 + 81 + … 625 ft 375 ft 225 ft 135 ft 81 ft Jeff Bivin -- LZHS

  29. A Bouncing Ball Upward = 375 + 225 + 135 + 81 + … 625 ft 375 ft 225 ft 135 ft 81 ft Jeff Bivin -- LZHS

  30. A Bouncing Ball Downward = 625 + 375 + 225 + 135 + 81 + … = 1562.5 Upward = 375 + 225 + 135 + 81 + … = 937.5 TOTAL = 2500 ft. 625 ft 375 ft 225 ft 135 ft 81 ft Jeff Bivin -- LZHS

  31. Find the sum of the series Jeff Bivin -- LZHS

  32. Fractions - Decimals Jeff Bivin -- LZHS

  33. Let’s try again + + Jeff Bivin -- LZHS

  34. One more subtract Jeff Bivin -- LZHS

  35. OK now a series Jeff Bivin -- LZHS

  36. .9 = 1 .9 = 1 That’s All Folks Jeff Bivin -- LZHS

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