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Sequence an ordered list of numbers Finding Patterns

Sequence an ordered list of numbers Finding Patterns. Arithmetic Geometric. Arithmetic Sequence. Pattern adding a fixed number from one term to the next COMMON DIFFERENCE d. a n =. NOTE n increases by only 1 at a time. n. Examples

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Sequence an ordered list of numbers Finding Patterns

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  1. Sequencean ordered list of numbersFinding Patterns Arithmetic Geometric

  2. Arithmetic Sequence Pattern adding a fixed number from one term to the next COMMON DIFFERENCE d

  3. an = NOTEn increases by only 1 at a time n

  4. Examples Sequence A: 5 , 8 , 11 , 14 , 17 , ... Sequence B: 26 , 31 , 36 , 41 , 46 , ... Sequence C: 20 , 18 , 16 , 14 , 12 , ...

  5. Common Difference - the fixed numbers that binds the sequence together In Sequence A the common difference is +3 In Sequence B the common difference is +5 In Sequence C the common difference is -2

  6. Common Difference = d Generic sequence is referred to using the letter a along with subscripts as follows: Generic Sequence: a1, a2, a3, a4, ... The fifth term of a given sequence = a5. • The 17th term = a17 • The nth term = an The term right before the nth term = an-1 d can be calculated by subtracting any two consecutive terms in an arithmetic sequence. d = an - an - 1, where n is any positive integer greater than 1.

  7. Common difference (d) = 3 an = dn + c or an= 3n + c c is some number that must be found

  8. Common Difference = 5 Formula for the nth term = an = 5n + 21 14th term a14 = 5(14) + 21 = 70 + 21 = 91 40th term a40= 5(40) + 21 = 200 + 21 = 221

  9. Common difference = -2. Formula will be -2n + c Find c -2×1 + c = -2 + c-2×2 + c = -4 + c-2×3 + c = -6 + c-2×4 + c = -8 + c-2×5 + c = -10 + c C = 22

  10. Geometric Sequence Pattern Multiply a fixed number from one term to the next Common Ratio r

  11. an= NOTEn increases by only 1 at a time n

  12. a1 a2 a3 a4 5, 10, 20, 40, ... multiply each term by 2 to arrive at the next term or...dividea2 by a1 to find the common ratio, 2 r = 2

  13. -11, 22, -44, 88, ... multiply each term by -2 to arrive at the next term or...dividea2 by a1 to find the common ratio, -2 r = -2

  14. Find the common ratio for the sequence r = divide the second term by the first term = -1/2. Checking shows that multiplying each entry by -1/2 yields the next entry.

  15. Find the 7th term of the sequence2, 6, 18, 54, ... n = 7; a1 = 2, r = 3 The seventh term is 1458.

  16. Find the 11th term of the sequence n = 11; a1 = 1, r = -1/2

  17. A ball is dropped from a height of 8’. The ball bounces to 80% of its previous height with each bounce. How high (to the nearest tenth of a foot) does the ball bounce on the fifth bounce?

  18. Set up a model drawing for each "bounce". 6.4, 5.12, ___, ___, ___ The common ratio is 0.8. Answer: 2.6 feet

  19. Finding More Patterns Fractal

  20. Carl Friedrich Gauss Germany (1777 - 1855)

  21. Add the integers from 1 to 100

  22. Add the integers from 1 to 100 There are 50 pairs of 101...

  23. Do you think we can find a formula that will work for adding all the integers from 1 to n?

  24. Do you think we can find a formula that will work for adding all the integers from 1 to n? • How many pairs of n + 1 are there? Half of n! 5050 + 101 = 5151

  25. Write this series in sigma notation?

  26. To : A similar formula works for when the terms skip some numbers, like Find the sum of the first n terms

  27. This is for arithmetic sequences ONLY! • Let's find the sum of the first 50 terms of the arithmetic sequence • We have: • We need:

  28. Finding More Patterns Fractal

  29. Start with an equilateral triangle. To each side…. What is its perimeter? Koch’s Snowflake Fractal Pattern

  30. Divide each side into thirds… The side length of each successive small triangle is 1/3 What is its perimeter? Koch’s Snowflake Fractal Pattern

  31. Total length increases by one third and thus the length at step n will be (4/3)n Koch’s Snowflake Fractal

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