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Sequences finding a rule

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Sequences finding a rule

  1. 1. 1
  2. 2. A sequence is a set of terms, in a definite order, where the terms are obtained by some rule. A finite sequence ends after a certain number of terms. An infinite sequence is one that continues indefinitely.
  3. 3. For example: 1, 3, 5, 7, … (This is a sequence of odd numbers) 1st term = 2 x 1 – 1 = 1 2nd term = 2 x 2 – 1 = 3 3rd term = 2 x 3 – 1 = 5 nth term = 2 x n – 1 = 2n - 1 . . . . . . + 2 + 2
  4. 4. NOTATION 1st term = u 2nd term = u 3rd term = u nth term = u . . . . . . 1 2 3 n
  5. 5. OR 1st term = u 2nd term = u 3rd term = u nth term = u . . . . . . 0 1 2 n-1
  6. 6. FINDING THE FORMULA FOR THE TERMS OF A SEQUENCE
  7. 7. A recurrence relation defines the first term(s) in the sequence and the relation between successive terms.
  8. 8. u = 5 u = u +3 = 8 u = u +3 = 11 u = u +3 = 3n + 2 . . . 1 2 3 n+1 For example: 5, 8, 11, 14, … 1 2 n
  9. 9. What to look for when looking for the rule defining a sequence
  10. 10. Constant difference: coefficient of n is the difference 2nd level difference: compare with square numbers (n = 1, 4, 9, 16, …) 3rd level difference: compare with cube numbers (n = 1, 8, 27, 64, …) None of these helpful: look for powers of numbers (2 = 1, 2, 4, 8, …) Signs alternate: use (-1) and (-1) -1 when k is odd +1 when k is even kk 2 3 n - 1
  11. 11. EXAMPLE: Find the next three terms in the sequence 5, 8, 11, 14, …
  12. 12. EXAMPLE: The nth term of a sequence is given by x = a) Find the first four terms of the sequence. b) Which term in the sequence is ? c) Express the sequence as a recurrence relation. 1__ 2 nn 1 1024 ____
  13. 13. EXAMPLE: Find the nth term of the sequence +1, -4, +9, -16, +25, …
  14. 14. EXAMPLE: A sequence is defined by a recurrence relation of the form: M = aM + b. Given that M = 10, M = 20, M = 24, find the value of a and the value of b and hence find M . n + 1 1 32 4
  15. 15. This powerpoint was kindly donated to www.worldofteaching.com http://www.worldofteaching.com is home to over a thousand powerpoints submitted by teachers. This is a completely free site and requires no registration. Please visit and I hope it will help in your teaching.
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