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Objective: 1. After completing activity 1, mod. 10 2. With 90% accuracy

Learn to identify, write formulas, and find the number of terms in arithmetic and geometric sequences. Also, explore finite series and find their sums.

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Objective: 1. After completing activity 1, mod. 10 2. With 90% accuracy

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  1. Objective: 1. After completing activity 1, mod. 10 2. With 90% accuracy 3. -Identify sequences as arithmetic, geometric, or neither • -Write recursive formulas for arithmetic and geometric sequences • -Write explicit formulas for arithmetic and geometric sequences • -Determine the number of terms in a finite arithmetic or geometric sequence

  2. Arithmetic Sequence: • An arithmetic (linear) sequence is a sequence of numbers in which each new term ( )is calculated by adding a constant vale(d) to the previous term. • For example: 1,2,3,4,5,6,… The value of d is 1. Find the constant value that is added to get the following sequences & write out the next 3 terms. • 2,6,10,14,18,22,…

  3. Simple test to check if a pattern is an arithmetic sequence: • Check that the difference between consecutive terms is constant. • For example, in the sequence: 1,2,3,4,5,6,… the constant is one because 6-5=5-4=4-3=3-2=2-1=1 In other words, since the difference is constantly 1, then it is an arithmetic sequence

  4. Find the constant value that is added to get the following sequences & write out the next 3 terms • 2,6,10,14,18,22,… (you did this one already) • -5,-3,-1,1,3,… • 1,4,7,10,13,16,… • -1,10,21,32,43,54,… • 3,0,3,-6,-9,-12,…

  5. The recursive formula for an arithmetic sequence For example, the recursive formula for the arithmetic sequence 1,2,3,4,5,6… is

  6. Write the recursive formula for each sequence: • 2,6,10,14,18,22,… • -5,-3,-1,1,3,… • 1,4,7,10,13,16,… • -1,10,21,32,43,54,… • 3,0,3,-6,-9,-12,…

  7. Answers:

  8. Explicit formula • The explicit formula for a sequence defines any term based on its term number (n):

  9. Write the explicit formula for each sequence • 2,6,10,14,18,22,… • -5,-3,-1,1,3,… • 1,4,7,10,13,16,… • -1,10,21,32,43,54,… • 3,0,3,-6,-9,-12,…

  10. Answers:

  11. Use your explicit formulas to answer the questions: (show your work) 1. What is the third term in the pattern: 2,6,10,14,18,22,… 2.What is the 20th term? 3.What is the 35th term?

  12. Answers:

  13. Geometric Sequence: • A geometric sequence- every term after the first is formed by multiplying the preceding term by a constant value called the common ratio (or r) • For example: 2,10,50,250,1250 • The value of r is 5 because :

  14. Simple test to check if a sequence is a geometric sequence: When you divide a term by a previous term you must arrive at equal common ratios.

  15. Determine the common factor for the following geometric sequence: • 5,10,20,40,80,… • 7,28,112,448,… • 2,6,18,54,…

  16. Answer: • 2 • ½ • 4 • 3

  17. The recursive formula for a geometric sequence • Write the recursive formula for each geometric sequence : • 5,10,20,40,80,… • 7,28,112,448,… • 2,6,18,54,…

  18. Write the recursive formula for each geometric sequence • 5,10,20,40,80,… • 7,28,112,448,… • 2,6,18,54,…

  19. Answers:

  20. The explicit formula for a geometric series is: • Write the explicit formula for each geometric sequence : • 5,10,20,40,80,… • 7,28,112,448,… • 2,6,18,54,…

  21. Answers:

  22. Is the following sequence arithmetic or geometric?: -3,30,-300,3000,…. • Write a recursive & explicit formula for it. • Use the explicit formula to find the 8th term.

  23. Answer: • Geometric

  24. Warm-up 3/8/10 • State whether the sequence is arithmetic, geometric, or neither. Use your notes.

  25. Answers: • Arithmetic • Neither • Geometric

  26. Warm-Up 3/9/10 (head your paper) • Consider the following arithmetic sequences: • 0, 6, 12, 18,…120 • 1, 9, 17, 25,…97 • 15, 12, 9, 6,…-21 What is the common difference for each? How many terms are in the sequence?

  27. Class work: • Remember to head your notebook with page # & today’s date. • Old green Alg. II Book page 476 #1-9 under Exercises & Applications also do #13 & 14 . Be Ready for a Quiz on it! • Your regular book: Page 277 Assignment #1.1, 1.3, 1.4, 1.9

  28. Class work: 3/9/10 • Old green Alg. II book page 479 #35-37 • Read the problem carefully • Your regular book Page 277 Assignment #1.5 a-c

  29. Homework: • Page 274-275 a-f. • Page 277 Warm-up #1-2

  30. Homework Review, Check your work: • Page 274 Discussion a. • 1. arithmetic • 2. geometric • 3. neither • 4. geometric • 5. neither • 6. Fibonacci

  31. Discussion b.

  32. Objective: 1.After completing activity 2, mod. 10 2. With 90% accuracy 3.-Identify sequences as arithmetic, geometric, or neither • Write explicit formulas for arithmetic and geometric sequences • Determine the number of terms in a finite arithmetic sequence • Write formulas for finite arithmetic series

  33. Activity 2 Notes • Finite Series- the sum ( )of the terms of a finite sequence. • For example: a finite series with n terms is: Arithmetic Series: the sum of the terms of an arithmetic sequence. For example:

  34. Find the sum of the first 100 natural numbers:

  35. Activity 2 Notes • Class work: page 281 exploration • Parts a-c only

  36. Answers to Exploration:

  37. Conclusion Question:

  38. Formula:

  39. Formula 2: • The sum of the terms of a finite arithmetic sequence with n terms & a common difference d can also be found by using the formula: • Please notice that this formula involves the common difference d.

  40. Warm-up: 3/11/10 • Consider the sequence: • 7,11,15,…59 • Find the sum of all the terms • Answer: 462

  41. Homework: 3/10/10 • Warm –up page 282-283 #1-3 • Check your HW: 1. 2,001,000 2a. )3 2b.) 4 2c.) 113 2d. ) 25,561 3a.) 780 3b.) 1197

  42. Class work: 3/11/10 • Assignment page 283-284 # 2.1, 2.3, 2.4, 2.5, 2.6, 2.7

  43. Answers to Assignment: • 2.1) sum of first n even numbers: n(n+1) • 2.3) No because the sum of each pair is not a constant • 2.4a.) the monthly payments can be considered to be an arithmetic sequence where the first term is $206.26 and the common difference is $206.26 • 2.4B) Yes. The payments form an arithmetic sequence, their sum forms a series.

  44. Answers to Assignment: 2.4c) The lessee pays 2.4d) The difference of $2535.64 may be the cost to purchase the car at the end of the lease.

  45. Answers to Assignment: • 2.5) • 2.6a) -2,3,8 • 2.6b) 743 • 2.6c) 55,575

  46. Answers to Assignment: • 2.7a) 2562.5 • 2.7b) 2.875 • 2.7c) 7.875 and 10.75

  47. Objective 1.After completing activity 3, mod. 10 2.With 90% accuracy 3.Identify sequences as arithmetic, geometric, or neither • Write explicit formulas for arithmetic and geometric sequences • Determine the number of terms in a finite geometric sequence • Write formulas for finite geometric series

  48. Geometric Series: Geometric Series- the sum of the terms of a geometric sequence. For example: 2,6,18,54,162

  49. Explore: • Head your notebook with today’s date, page # & title. • With your partner, try the exploration on page 284-285 parts a-g

  50. Formula: • The sum of a finite geometric series with n terms and a common ratio r: • Use the formula with the geometric sequence: 2,6,18,54,162 to find the sum of all 5 terms.

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