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  # 9.4 part 1 and 2 combined worked
  
  Sequences
  
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  ### 9.4 part 1 and 2 combined worked
  
  1. 1. Homework Questions
  2. 2. Number Patterns  Find the next two terms, state a rule to describe the pattern. 1. 1, 3, 5, 7, 9… 2. 16, 32, 64… 3. 50, 45, 40, 35… 4. -3, -7, -11, -15…
  3. 3. Sequence Notation  A sequence is an ordered list of numbers – each number is a term.  State the first 5 terms:  an = n  (plug in 1, 2, 3, 4, 5)  1, 2, 3, 4, 5
  4. 4. More Examples 1. an = 4n 2. an = 2n-3 3. an = |1-n2 | 4. an = 5. an = 3 1 n − 3 6n
  5. 5. Recursive v. Explicit
  6. 6. Definition  Recursive Formula – a sequence is recursively defined if the first term is given and there is a method of determining the nth tem by using the terms that precede it.  English – if you can use the term before it to figure out what comes next  Ex: {-7, -4, -1, 2, 5, …} 
  7. 7. Examples of Recursive  {-9, -4, -2, 0, 2, …}  {-4, -8, -16, -32, -64, …}  {6, 11, 16, 21, 26, …}  {8, 4, 2, 1, …}
  8. 8. Definition  Explicit Formula – a formula that allows direct computation for any term for a sequence  English – you don’t need to term prior in order to figure out what the nth term is going to be.  Ex: {8, 9, 10, 11, 12, …}  an= n + 7
  9. 9. Examples of Explicit  {-3, 1, 5, 9, …}  {1, 4, 9, 16, …}  {7, 9, 11, 13, …}  {24, 20, 16, 12, …}
  10. 10. Arithmetic Sequences
  11. 11. Arithmetic Sequences  In an arithmetic sequence, the difference between consecutive terms is constant.  The difference is called the common difference.  To find d: 2nd term – 1st term
  12. 12. Arithmetic? 1. 2, 4, 8, 16 2. 6, 12, 18 3. 48, 45, 42 4. 2, 5, 7, 12
  13. 13. Arithmetic Sequence Formulas  Recursive Formula  an = an-1 + d  use if you know prior terms  Explicit Formula  an = a1 + (n-1)d  an = nth term  a1 = 1st term  n = number of terms  d = common difference
  14. 14. Examples  Find the 20th term of each sequence 1. 213, 201, 189, 177… 2. .0023, .0025, .0027…
  15. 15. More examples  Find the 17th term of the sequence: 3. a16 = 18, d = 5
  16. 16. Find the missing term  Use arithmetic mean = average! 4. 84, _______, 110 5. 24, _______, 57
  17. 17. Geometric Sequences
  18. 18. Geometric Sequences  In a geometric sequence, the ratio between consecutive terms is constant.  This ratio is called the common ratio.  To find r: stterm ndterm 1 2
  19. 19. Geometric, Arithmetic, Neither? (find the next 2 terms if so) 1. 5, 15, 45, 135… 2. 15, 30, 45, 60… 3. 6, -24, 96, -384… 4. 8, 20, 32, 44…
  20. 20. Geometric Sequences Formulas  Recursive Formula  an = an-1 r  Explicit Formula  an = a1 rn-1 • •
  21. 21. Find the 19th term… 1. 11, 33, 99, 297… 2. 20, 17, 14, 11, 8…
  22. 22. FYI - Graphs  Arithmetic Graphs are linear  Geometric Graphs are exponential
  23. 23. Geometric Mean  Geometric Mean = 3. 20, _____, 80 4. 3, ____, 18, 75 5. 28, ____, 5103 numbersproductof 2
  24. 24. Homework  WORKSHEET!  We need to talk about numbers 16-20 though, so wait on me!
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