1. Section 8.1
Sequences

2. Back to Work…

3. Section 8.1 Sequences Learning Targets:
I can identify and use arithmetic and geometric sequences
I can graph a sequence
I can find the limits of sequences (arithmetic and geometric)

4. Defining a Sequence
A sequence an is a list of numbers written in an explicit order. If the domain is finite, then the sequence is a finite sequence. If the domain is an infinite subset of positive integers, it is an infinite sequence.

5. Example 1
Find the first six terms and the 100th term of the sequence:

6. Definitions An explicit formula is defined in terms of n.
A recursive formula relates each new term to a previous term.

7. Example 2
Find the first four terms and the eighth term for the sequence defined recursively by the conditions: for all n 2
2 options…
Keep adding until the 8th term.
Write an explicit definition to find the 8th term (this isn’t ALWAYS an option).

8. Arithmetic and Geometric Sequences

9. Example 3
For each of the following arithmetic sequences, find (a) the common difference, (b) a recursive rule for the nth term, (c) an explicit rule for the nth term, and (d) the ninth term.
Sequence 1: -5, -2, 1, 4, 7, … Sequence 2: ln 2, ln 6, ln 18, ln 54, …

10. Definition

11. Example 4
For each of the following geometric sequences, find (a) the common ratio, (b) a recursive rule for the nth term, (c) an explicit rule for the nth term, and (d) the tenth term.
Sequence 1: 1, -2, 4, -8, 16, … Sequence 2: 10-2, 10-1, 1, 10, 102, …

12. 8.1A Homework
#5, 9, 11, 15, 19, 21

13. Please have your homework out so I can come around to check it off!

14. Example 5
The second and fifth terms of a geometric sequence are 6 and -48 respectively. Find the first term, the common ratio, and an explicit rule for the nth term.
-48 6

15. Limit of a Sequence
Definition: We write and say that the sequence converges to L. Sequences that do not have limits diverge.
Sum Rule:
Difference Rule:
Product Rule:
Constant Multiple Rule:
Quotient Rule

16. Therefore, the sequence CONVERGES.
Example 6
Determine whether the sequence converges or diverges. If it converges, find its limit.
Therefore, the sequence CONVERGES.

17. Example 7 2 limits, therefore DIVERGES Arithmetic… so always DIVERGES
Determine whether the sequence with given nth term converges or diverges. If it converges, find the limit.
(a) n=1, 2, …
(b) For all n 2
2 limits, therefore DIVERGES
Arithmetic… so always DIVERGES

18. Sandwich (“Squeeze”) Theorem
Min value of cos n = -1
Max value of cos n = 1

19. Absolute Value Theorem

20. 8.1B Homework
#31 – 37 odd, 41,