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SEQUENCES

Learn about sequences: finite vs. infinite, notation for terms, finding formulas, rules for defining sequences, and solving sequence problems with examples. Gain insights into different types of sequences and methods for determining next terms. Enhance your understanding through clear explanations and practice exercises.

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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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