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Understanding Number Sequences: Formulas, Patterns & Fibonacci Sequence

Learn about number sequences, evaluate terms using formulas, find explicit formulas for nth terms, use recursion formulas, and explore patterns like the Fibonacci Sequence. Practice writing terms and finding sums.

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Understanding Number Sequences: Formulas, Patterns & Fibonacci Sequence

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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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