1. Unit 7: Sequences and Series

2. Sequences A sequence is a list of #s in a particular order
If the sequence of numbers does not end, then it is called an infinite sequence
Each # in a sequence is called a term
Ex. 3, 5, 7, 9….
a1=3
a2=5

3. Arithmetic Sequences
A sequence in which each term after the first term is found by adding a constant (called the common difference (d)), to the previous term
Ex. Find the common difference (one term minus the previous term)
55, 49, 43, 37, 31, 25,19

4. Arithmetic Sequences
Formula for the general pattern for any arithmetic sequence:

5. Arithmetic Sequences
Write a equation for the nth term of the following sequence

6. Arithmetic Sequences
Write a equation for the nth term of the following sequence

7. Arithmetic Sequences
Find a15

8. Arithmetic Sequences
In the sequence below, which term has a value of 286?

9. Arithmetic Sequences
What is the value of the first term if the 9th and 10th terms are 4 and 2 consecutively?

10. Arithmetic Sequences
If the 3rd term of an arithmetic sequence is 8 and the 16th term is 47, find a1 and d

11. Arithmetic Means
Terms between any two non-successive terms of an arithmetic sequence
Find 4 arithmetic means between 16 and 91

12. Arithmetic Means
Find 1 arithmetic mean between 50 and -120

13. Results-4A
Concerns: arithmetic means, notation, 2x problem, more practice

14. Arithmetic Series
A series is the indicated sum of the terms of a sequence
If the seq. is 18, 22, 26, 30
The arith series is

15. Arithmetic Series
To find the sum of an arithmetic series,

16. Arithmetic Series
Find the sum of the first 100 positive integers.

17. Arithmetic Series
Find the sum of the first 20 even numbers beginning with 2.

18. Arithmetic Series
Find the sum of …+2

19. Arithmetic Series
Find the sum of …+97

20. Geometric Sequences
A geometric sequence is a sequence where each # in the seq. is multiplied by a constant (which is called the common ratio (r))
To find r, divide any term in the sequence by the previous term

21. Geometric Sequences
General Formula:

22. Geometric Sequences
Find the 11th term of the geo. Sequence listed below
64, -32, 16, -8,…a11

23. Geometric Sequences
Find the 6th term of the geo. sequence listed below
3, -15, 75, ..a6

24. Geometric Sequences Write an equation for the nth term

25. Geometric Sequences
Find the 10th term of the sequence if
a4=108
r=3

26. Geometric Sequences
Find the 7th term of the sequence if
a3=96
r=2

27. Geometric Means
Geometric means are the missing terms between two non-successive terms in a geo. Sequence
Find 3 geometric means between 2.25 and 576

28. Geometric Means
Find 5 geometric means between ½ and 1/1458

29. Geometric Series
A series that is associated with a geometric sequence

30. Geometric Series
Find the sum of the first 6 terms of the geometric series …

31. Geometric Series
What is r if the sum of the first 6 terms in a geo series is 11,718 and the first term is 3
Hint: solve by doing an intersection on the graph

32. Geometric Series
Find the first term of the series if the S8=39,360 and r=3

33. Geometric Series
Find the sum of the first 8 terms of
1+x+x2+x3+…

34. Geometric Series Find the a1 if Sn=165, an=48, r=-2/3 Hint: an=a1.rn-1
So, r.an=a1.rn-1.r
an.r=a1.rn (now substitute into the sum formula)

35. Sigma Notation
More concise (less time consuming) notation for writing out a series

36. Sigma Notation

37. Sigma Notation

38. Sigma Notation

39. Write in sigma notation

40. Write in sigma notation

41. Write in sigma notation

42. Write in sigma notation

43. Write in sigma notation

44. Infinite Geometric Series
In an infinite series, Sn approaches some limit as n becomes very large. That limit is defined to be the sum of the series. If an infinite series has a sum, it is said to converge.
A series converges (or has a sum) if and only if lrl < 1

45. Does the geom. series have a sum?

46. Does the geom. series have a sum?

47. Does the sum of each term approach some limit?

48. To find the sum of an infinite series
Make sure a limit exists first

49. An infinite series in sigma notation—find the sum

50. An infinite series in sigma notation—find the sum

51. Pascal’s Triangle
*First row is used for anything to the zero power **used for the coefficients of each term of the expanded binomial

52. Expand using Pascal’s Triangle

53. Expand using Pascal’s Triangle

54. Writing a repeating decimal as a fraction

55. Writing a repeating decimal as a fraction