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SECTION 7.1

SECTION 7.1. SEQUENCES. SEQUENCES. A number sequence is an arrangement of numbers in which there is first number, a second number, a third number, . . . Example: 2 , 4 , 6 , 8 , . . . We call this an infinite sequence. NOTATION:. In general, a number sequence is denoted as:

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SECTION 7.1

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  1. SECTION 7.1 SEQUENCES

  2. SEQUENCES A number sequence is an arrangement of numbers in which there is first number, a second number, a third number, . . . Example: 2 , 4 , 6 , 8 , . . . We call this an infinite sequence.

  3. NOTATION: In general, a number sequence is denoted as: a 1 , a 2 , a 3 , a 4 , a 5 , . . . The subscript denotes the position of the term in the sequence.

  4. NOTATION: A general term in the sequence is denoted as: a n An entire general sequence might be denoted as: { a n }

  5. EVALUATING TERMS IN A SEQUENCE: A sequence { pn } is determined by the formula pn = n 2 - n + 1. Evaluate p 3 and p 10 p 3 = 3 2 - 3 + 1 = 7 p 10 = 10 2 - 10 + 1 = 91

  6. WRITING THE FIRST SEVERAL TERMS OF A SEQUENCE Write down the first six terms of the sequence: a5 =4/5 a6 = 5/6 a1 = 0 a2 = 1/2 a3 = 2/3 a4 = 3/4

  7. USING THE GRAPHING CALCULATOR Find the first six terms of the sequence: 2nd List Ops Choose Seq seq(expression,variable,begin,end, increment)

  8. WRITING THE FIRST SEVERAL TERMS OF A SEQUENCE Write down the first six terms of the sequence: b5 = 2/5 b6 = -1/3 b1 = 2 b2 = - 1 b3 = 2/3 b4 = -1/2

  9. FINDING AN EXPLICIT FORMULA FOR NTH TERM a 1 = 2 = 2(1) a 2 = 4 = 2(2) a 3 = 6 = 2(3) a 4 = 8 = 2(4) a n = 2(n) Thus, if we need to know the value of the 39th term: a 39 = 2(39) = 78

  10. FINDING AN EXPLICIT FORMULA FOR NTH TERM a 1 = 0 = (1-1)2 a 2 = 1 = (2-1)2 a 3 = 4 = (3-1)2 a 4 = 9 = (4-1)2 a 5 = 16 = (5-1)2 Give an explicit formula for the sequence { a n } with initial terms: 0, 1, 4, 9, 16, 25, . . . a n = (n - 1)2

  11. Use the explicit formula for { a n } to determine the value of a 13 a n = (n - 1)2 a 13 = (13 - 1)2 a 13 = 144

  12. THE FACTORIAL SYMBOL If n > 0 is an integer, the factorial symbol n! is defined as follows: 0! = 1 1! = 1 n! = n(n - 1) · . . . · 3 · 2 · 1 if n > 2 n! = n(n - 1)!

  13. SEQUENCE FORMULAS Seeing Patterns: b) 1, 3, 6, 10, 15, 21, 28, . . . b1 = 1 bn = bn - 1 + n Recursion Formula

  14. EXAMPLE Write down the first five terms of the sequence defined by: s1 = 1 sn = 4s n - 1 s4 = 4 · 16 = 64 s5 = 4 · 64 = 256 s2 = 4 · 1 = 4 s3 = 4 · 4 = 16

  15. EXAMPLE: Find the pattern in 1, 1, 2, 3, 5, 8, 13, . . . f1 = f2 = 1 and fn = fn - 2 + fn - 1 This is called the Fibonacci Sequence

  16. EXAMPLE: Determine the first six terms of the sequence:

  17. EXAMPLE: Determine the first six terms of the sequence: b1 = 0 b2 = 1

  18. Sums are usually denoted in sigma notation: am + am + 1 + am + 2 + . . . + an a 3 + a 4 + a 5 + a 6

  19. 1 + 2 2 + 3 2 + . . . + 100 2

  20. EXAMPLE We have no formula for this but, with patience, could find the sum with a calculator.

  21. CONCLUSION OF SECTION 7.1

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