1. Introduction to Arithmetic Sequences 18 May 2011

2. Arithmetic Sequences When the difference between any two numbers is the same constant value This difference is called d or the constant difference {4, 5, 7, 10, 14, 19, …} {7, 11, 15, 19, 23,...} ← Not an Arithmetic Sequence ← Arithmetic Sequence d = 4

3. Your Turn: Determine if the following sequences are arithmetic sequences. If so, find d (the constant difference). {14, 10, 6, 2, –2, …} {3, 5, 8, 12, 17, …} {33, 27, 21, 16, 11,…} {4, 10, 16, 22, 28, …}

4. Recursive Form The recursive form of a sequence tell you the relationship between any two sequential (in order) terms. u n = u n–1 + d n ≥ 2 common difference

5. Writing Arithmetic Sequences in Recursive Form If given a term and d 1. Substitute d into the recursive formula

6. Examples: Write the recursive form and find the next 3 terms u 1 = 39, d = 5

7. Your Turn: Write the recursive form and find the next 3 terms u 1 = 8, d = –2 u 1 = –9.2, d = 0.9

8. Writing Arithmetic Sequences in Recursive Form, cont. If given two, non-sequential terms 1. Solve for d d = difference in the value of the terms difference in the number of terms 2. Substitute d into the recursive formula

9. Example #1 Find the recursive formula u 3 = 13 and u 7 = 37

10. Example #2 Find the recursive formula u 2 = –5 and u 7 = 30

11. Example #3 Find the recursive formula u 4 = –43 and u 6 = –61

12. Your Turn Find the recursive formula: 1. u 3 = 53 and u 5 = 712. u 2 = -7 and u 5 = 8 3. u 3 = 1 and u 7 = -43

13. Explicit Form The explicit form of a sequence tell you the relationship between the 1 st term and any other term. u n = u 1 + (n – 1)d n ≥ 1 common difference

14. Summary: Recursive Form vs. Explicit Form Recursive Form u n = u n–1 + d n ≥ 2 Sequential Terms Explicit Form u n = u 1 + (n – 1)d n ≥ 1 1 st Term and Any Other Term

15. Writing Arithmetic Sequences in Explicit Form You need to know u 1 and d!!! Substitute the values into the explicit formula 1. u 1 = 5 and d = 22. u 1 = -4 and d = 5

16. Writing Arithmetic Sequences in Explicit Form, cont. You may need to solve for u 1 and/or d. 1. Solve for d if necessary 2. Back solve for u 1 using the explicit formula u 4 = 12 and d = 2

17. Example #2 u 7 = -8 and d = 3

18. Example #3 u 6 = 57 and u 10 = 93

19. Example #4 u 2 = -37 and u 7 = -22

20. Your Turn: Find the explicit formulas: 1. u 5 = -2 and d = -6 2. u 11 = 118 and d = 13 3. u 3 = 17 and u 8 = 92 4. u 2 = 77 and u 5 = -34

21. Using Explicit Form to Find Terms Just substitute values into the formula! u 1 = 5, d = 2, find u 5

22. Using Explicit Form to Find Terms, cont. u 1 = -4, d = 5, find u 10

23. Your Turn: 1. u 1 = 4, d = ¼2. u 1 = -6, d = ⅔ Find u 8 Find u 4 3. u 1 = 10, d = -½4. u 1 = π, d = 2 Find u 12 Find u 27

24. Summations Summation – the sum of the terms in a sequence {2, 4, 6, 8} → 2 + 4 + 6 + 8 = 20 Represented by a capital Sigma

25. Summation Notation Sigma (Summation Symbol) Upper Bound (Ending Term #) Lower Bound (Starting Term #) Sequence

26. Example #1

27. Example #2

28. Example #3

29. Your Turn: Find the sum:

32. Partial Sums of Arithmetic Sequences – Formula #1 Good to use when you know the 1 st term AND the last term # of terms 1 st term last term

33. Formula #1 – Example #1 Find the partial sum: k = 9, u 1 = 6, u 9 = –24

34. Formula #1 – Example #2 Find the partial sum: k = 6, u 1 = – 4, u 6 = 14

35. Formula #1 – Example #3 Find the partial sum: k = 10, u 1 = 0, u 10 = 30

36. Your Turn: Find the partial sum: 1. k = 8, u 1 = 7, u 8 = 42 2. k = 5, u 1 = –21, u 5 = 11 3. k = 6, u 1 = 16, u 6 = –19

37. Partial Sums of Arithmetic Sequences – Formula #2 Good to use when you know the 1 st term, the # of terms AND the common difference # of terms 1 st termcommon difference

38. Formula #2 – Example #1 Find the partial sum: k = 12, u 1 = –8, d = 5

39. Formula #2 – Example #2 Find the partial sum: k = 6, u 1 = 2, d = 5

40. Formula #2 – Example #3 Find the partial sum: k = 7, u 1 = ¾, d = –½

41. Your Turn: Find the partial sum: 1. k = 4, u 1 = 39, d = 10 2. k = 5, u 1 = 22, d = 6 3. k = 7, u 1 = 6, d = 5

42. Choosing the Right Partial Sum Formula Do you have the last term or the constant difference?

43. Examples Identify the correct partial sum formula: 1. k = 6, u 1 = 10, d = –3 2. k = 12, u 1 = 4, u 12 = 100

44. Your Turn: Identify the correct partial sum formula and solve for the partial sum 1. k = 11, u 1 = 10, d = 2 2. k = 10, u 1 = 4, u 10 = 22 3. k = 16, u 1 = 20, d = 7 4. k = 15, u 1 = 20, d = 10 5. k = 13, u 1 = –18, u 13 = –102