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SEQUENCES

SEQUENCES. 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. 1, 3, 5, 7, … (This is a sequence of odd numbers).

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SEQUENCES

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  1. SEQUENCES

  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. 1, 3, 5, 7, … (This is a sequence of odd numbers) 1st term = 2 x 1 – 1 = 1 For example: + 2 2nd term = 2 x 2 – 1 = 3 + 2 3rd term = 2 x 3 – 1 = 5 . . . . . . nth term = 2 x n – 1 = 2n - 1

  4. NOTATION 1 2nd term = u 1st term = u 2 3rd term = u 3 . . . . . . nth term = u n

  5. OR 0 2nd term = u 1st term = u 1 3rd term = u 2 . . . . . . nth term = u n-1

  6. FINDINGTHE FORMULA FORTHE TERMS OFA SEQUENCE

  7. A recurrence relation defines the first term(s) in the sequence and the relation between successive terms.

  8. For example: 5, 8, 11, 14, … 1 u = u +3 = 8 u = 5 1 2 u = u +3 = 11 3 2 . . . u = u +3 = 3n + 2 n+1 n

  9. What to look for when looking for the rule defining a sequence

  10. 2 • 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 3 n - 1 k k

  11. EXAMPLE: Find the next three terms in the sequence 5, 8, 11, 14, …

  12. 1 __ n n 2 EXAMPLE: The nth term of a sequence is given by x = • Find the first four terms of the sequence. b) Which term in the sequence is ? c) Express the sequence as a recurrence relation. 1 ____ 1024

  13. EXAMPLE: Find the nth term of the sequence +1, -4, +9, -16, +25, …

  14. n + 1 1 2 3 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 . 4

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