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Sequences

May 10-13 and May 17-20: School starts at 7:15 for EOCT testing!. Sequences. EOCT: May 10-11. Vocabulary. Sequence: an ordered list of numbers Ex. -2, -1, 0, 1, 2, 3 Term: each number in a sequence Ex. a 1 , a 2 , a 3 , a 4 , a 5 , a 6

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Sequences

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  1. May 10-13 and May 17-20: School starts at 7:15 for EOCT testing! Sequences EOCT: May 10-11

  2. Vocabulary • Sequence: an ordered list of numbers • Ex. -2, -1, 0, 1, 2, 3 • Term: each number in a sequence • Ex. a1, a2, a3, a4, a5, a6 • Infinite Sequence: sequence that continues infinitely • Ex: 2, 4, 6, 8, … • Finite Sequence: sequence that ends • Ex: 2, 4, 6 • Explicit Formula: defines the nth term of a sequence.

  3. Example 1: • Write the first six terms of the sequence defined by an = 4n + 5 • Write the first six terms of the sequence defined by an = 2n2 – 1

  4. Vocabulary • Recursive Formula: • Uses one or more previous terms to generate the next term. an-1

  5. Example 2: A) Write the first six terms of the sequence where a1 = -2 and an = 2an-1 – 1 B) Write the first six terms of the sequence where a1 = 4 and an = 3an-1 + 5

  6. May 10-13 and May 17-20: School starts at 7:15 for EOCT testing! Arithmetic Sequences EOCT: May 10-11

  7. Vocabulary • Arithmetic Sequence: • A sequence generated by adding “d” a constant number to pervious term to obtain the next term. • This number is called the common difference. • What is d? a2 – a1 • 3, 7, 11, 15, … d = 4 • 8, 2, -4, -10, … d =-6

  8. Formula for the nth term Common difference an = a1 + (n – 1)d First term in the sequence What term you are looking for What term you are looking for

  9. Example 1: • Find the 10th term of a1 = 7 and an = an-1 + 6 • Find the 7th term of a1 = 2.5 and an = an-1 - 3 d

  10. Example 2: • Find the 10th term of the arithmetic sequence where a3 = -5 and a6 = 16 • Find the 15th term of the arithmetic sequence where a5 = 7 and a10 = 22 • Find the 12th term of the arithmetic sequence where a3 = 8 and a7 = 20

  11. Vocabulary • Arithmetic Means: • Terms in between 2 nonconsecutive terms • Ex. 5, 11, 17, 23, 29  11, 17, 23 are the arithmetic means between 5 & 29

  12. Example 3: • Find the 4 arithmetic means between 10 & -30 • Find the 5 arithmetic means between 6 & 60

  13. May 10-13 and May 17-20: School starts at 7:15 for EOCT testing! Geometric Sequences EOCT: May 10-11

  14. Vocabulary • Geometric Sequence: • A sequence generated by multiplying a constant ratio to the previous term to obtain the next term. • This number is called the common ratio. • What is r? • 2, 4, 8, 16, … r = 2 • 27, 9, 3, 1, … r = 1/3

  15. Formula for the nth term First term in the sequence an = a1rn-1 What term you are looking for What term you are looking for Common Ratio

  16. Find the 5th term of a1 = 8 and an = 3an-1 Find the 7th term of a1 = 5 and an = 2an-1 Example 1

  17. Example 2: • Find a10 of the geometric sequence 12, 18, 27, 40.5, … • Find a7 of the geometric sequence where a1 = 6 and r = 4

  18. Homework P.140 #1-16 P.145 #1-17 ***Keep reviewing for your EOCT*** (May 10-11)

  19. Warm up 1. Find the 8th term of the sequence defined by a1= –4 and an= an-1+ 2 2. Find the 12th term of the arithmetic sequence in which a4= 2 and a7= 6 3. Find the four arithmetic means between 6 and 26. 4. Find the 5th term on the sequence defined by a1= 2 and an= 2an-1.

  20. May 10-13 and May 17-20: School starts at 7:15 for EOCT testing! Series(M2) EOCT: May 10-11

  21. Series • Series: the sum of a sequence • Sequence: 1, 2, 3, 4 • Series: 1 + 2 + 3 + 4 • Summation Notation: Summation Notation - __________________ EX. (for the above series)

  22. = _______ + _______ + _______ + _______ = ____ + _____ + _____ + _____ = _____

  23. Summation Properties • For sequences ak and bk and positive integer n: Not in packet!!

  24. Summation Formulas • For all positive integers n: ConstantLinear Quadratic

  25. Example 1: • Evaluate • Evaluate

  26. Extra Example:(Not in packet) • Evaluate Homework: P.135 #18-24 *work on Benchmark Practice WS*

  27. May 10-13 and May 17-20: School starts at 7:15 for EOCT testing! Arithmetic Series(M2) EOCT: May 10-11 Sequences and Series Test: May 18

  28. Vocabulary • An Arithmetic Series is the sum of an arithmetic sequence.

  29. Example 1: • Find the series 1, 3, 5, 7, 9, 11 B. Find the series 8, 13, 18, 23, 28, 33, 38

  30. Example 2: • Given 3 + 12 + 21 + 30 + …, find S25 • Given 16, 12, 8, 4, …, find S11

  31. Example 3: • Evaluate • Evaluate

  32. May 10-13 and May 17-20: School starts at 7:15 for EOCT testing! Geometric Series(M2) EOCT: May 10-11 Sequences and Series Test: May 18

  33. Vocabulary • An Geometric Series is the sum of an geometric sequence.

  34. Example 1: • Given the series 3 + 4.5 + 6.75 + 10.125 + …find S10 to the nearest tenth.

  35. Example 2: n • Evaluate • Evaluate r a1

  36. Homework • P. 141 #16-27 • P. 145 #18-23 • Study/Review for EOCT! (Sequences and Series ARE ON the EOCT)

  37. May 10-13 and May 17-20: School starts at 7:15 for EOCT testing! Infinite Geometric Series(M2) Sequences and Series Test: May 18 Finals: 1st Period – May 21 2nd Period – May 24 6th Period – May 26

  38. Vocabulary • An Infinite Geometric Series is a geometric series with infinite terms. SUM SUM

  39. Example 1: • Find the sum of the infinite geometric series 3 + 1.2 + 0.48 + 0.192 + … • Find the sum of the infinite geometric series 8 + 9.6 + 11.52 + 13.824 + …

  40. Example 2: • Find the sum of the infinite geometric series below:

  41. Example 3: NOT IN PACKET • Write 0.2 as a fraction in simplest form. • Write 0.04 as a fraction in simplest form.

  42. Homework • P. 147 #32 – 45 (M2 – Purple)

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