Understanding Arithmetic Sequences and Formulas

1. Exercise 1 + 2 + 3 + 4 + 5 + 6 21

2. Exercise 1 + 3 + 5 + 7 + 9 + 11 36

3. Exercise 2 + 4 + 6 + 8 + 10 + 12 42

4. Exercise 1 + 2 + 3 + … + 16 136

5. Sequence A sequence is a set of numbers, ordered according to a distinguishable pattern. The set can be either finite or infinite.

6. conjecture – generalized statement that seems to be true

7. ellipsis – shows pattern continues 1, 2, 3, 4, 5, 6, 7, …

8. Each number in a sequence is called a term.

9. nth term an

10. Position of the Term 1 2 3 4 5 6 … Term 18 3 6 9 12 15 …

11. Arithmetic Sequence A sequence of numbers, each differing by a constant amount from the preceding number, is an arithmetic sequence.

12. Common Difference The common difference, d, is the difference between successive terms of an arithmetic sequence.

13. Arithmetic Sequence Terms differ by a constant addend d. an = an–1 + d

14. Example 1 Find the value of d for the sequence 5, 9, 13, 17, 21, 25, 29, 33, … d = 4

15. Example 1 Find the value of a1for the sequence 5, 9, 13, 17, 21, 25, 29, 33, … a1 = 5

16. Example 1 Find the value of a5for the sequence 5, 9, 13, 17, 21, 25, 29, 33, … a5 = 21

17. Example 1 Find the value of a10for the sequence 5, 9, 13, 17, 21, 25, 29, 33, … a10 = 41

18. Example 2 Write the first six terms of the sequence in which a1 = 20 and d = –2. a1 = 20 a2 = 20 + (–2) = 18 a3 = 18 + (–2) = 16

19. Example 2 Write the first six terms of the sequence in which a1 = 20 and d = –2. a4 = 16 + (–2) = 14 a5 = 14 + (–2) = 12 a6 = 12 + (–2) = 10

20. Example 2 Write the first six terms of the sequence in which a1 = 20 and d = –2. The first six terms of the sequence are 20, 18, 16, 14, 12, and 10.

21. Example Is the sequence arithmetic? 4, –8, 16, –32, … no

22. Example Is the sequence arithmetic? 5, 15, 45, 135, … no

23. Example Is the sequence arithmetic? –4, –2, 0, 2, … yes

24. Example Is the sequence arithmetic? 20, 12, 4, –4, … yes

25. Arithmetic Sequence Terms differ by a constant addend d. an = an–1 + d

26. Recursive Formula A recursive formula specifies the step by which each term of the sequence is generated from the preceding term or terms.

27. Position of the Term 1 2 3 4 5 6 … Term 33 3 9 15 21 27 …

28. Example 3 Write the first five terms of the sequence defined by a1 = 2 and an= an – 1 + 9. a1 = 2 a2 = a1 + 9 = 2 + 9 = 11 a3 = a2 + 9 = 11 + 9 = 20

29. Example 3 Write the first five terms of the sequence defined by a1 = 2 and an= an – 1 + 9. a4 = a3 + 9 = 20 + 9 = 29 a5 = a4 + 9 = 29 + 9 = 38

30. Example 3 Write the first five terms of the sequence defined by a1 = 2 and an= an – 1 + 9. The first five terms of the sequence are 2, 11, 20, 29, and 38.

31. Example 4 Write the recursive formula for the sequence 28, 25, 22, 19, 16, … a1 = 28 d = –3 an = an – 1 – 3

32. Example Write the recursive formula for the sequence –4, –2, 0, 2, … a1 = – 4 an = an – 1 + 2

33. Example Write the recursive formula for the sequence 10, 7, 4, 1, … an = an – 1 – 3

34. 3, 9 ,15, 21, 27, 33 Add 6 to get the next term.

35. Position Term 1 3 3 + 1(6) = 9 2 3 + 2(6) = 15 3 3 + 3(6) = 21 4 3 + 4(6) = 27 5 3 + 5(6) = 33 6 3 + (n – 1)6 n

36. Explicit Formula The explicit formula for an arithmetic sequence is an = a1 + (n – 1)d, where n is the position in the sequence and d is the difference between terms.

37. Example 5 The explicit formula for the sequence 10, 15, 20, 25, 30, 35, … is an = 10 + (n – 1)5. Find a60. a60 = 10 + (60 – 1)5 = 10 + (59)5 = 10 + 295 = 305

38. Example 6 Write the explicit formula for the sequence –7, –4, –1, 2, 5, 8, … a1 = – 7; d = 3 an = a1 + (n – 1)d = – 7 + (n – 1)3 = – 7 + 3n – 3 = 3n – 10

39. Example 6 Use the simplified form of the explicit formula to find a72. an = 3n – 10 a72 = 3(72) – 10 = 216 – 10 = 206

40. Example Write the explicit formula for the sequence 20, 14, 8, 2, … an = 20 – (n – 1)6

41. Example Write the explicit formula for the sequence –10, –7, –4, –1, … an = –10 + (n – 1)3

42. Example Find a9 if an = 6 + (n – 1)2. 22

43. Example Find a12 if an = –4 + (n – 1)4. 40

44. Exercise If the house numbers on one side of a street form an arithmetic sequence where the first house number is 2, the second house number is 4, and the third house number is 6, how many houses are along that side of the street if the last house number is 38? 19

45. Exercise The first house number in a block in 13, and the twelfth house number is 90. Assuming the numbers form an arithmetic sequence, what is the number of the fourth house? 34

46. Exercise Locust Street has 16 lots for new houses. If the planners are laying out the numbers for the addresses of the new homes on that street and they are to form an arithmetic sequence, what is the largest common difference that could be used if the numbers cannot go above 50? 3