1. Geometric Sequences and Series

2. Arithmetic Series
Sum of Terms
Geometric Series
Sum of Terms
Arithmetic Sequences
Geometric Sequences
ADD
To get next term
MULTIPLY
To get next term

3. Geometric Sequence: sequence whose consecutive terms have a common ratio.
Example: 3, 6, 12, 24, 48, ...
The terms have a common ratio of 2.
The common ratio is the number r.
To find the common ratio you use an+1 ÷ an

4. Vocabulary of Sequences (Universal)

5. Find the next two terms of 2, 6, 18, ___, ___
6 – 2 vs. 18 – 6… not arithmetic
2, 6, 18, 54, 162
Find the next two terms of 80, 40, 20, ___, ___
40 – 80 vs. 20 – 40… not arithmetic
80, 40, 20, 10, 5

6. Find the next two terms of -15, 30, -60, ___, ___
30 – -15 vs. -60 – 30… not arithmetic
-15, 30, -60, 120, -240
Find the next three terms of 2, 3, 9/2, ___, ___, ___
3 – 2 vs. 9/2 – 3… not arithmetic

7. Find the 8th term if a1 = -3 and r = -2.NA -2

8. Find the 10th term if a4 = 108 and r = 3.?? an 10 NA 3

9. Write an equation for the nth term of the geometric sequence 3, 12, 48, 192, …

10. 6, 18, 54 are the Geometric Mean between 2 and 162
Geometric Mean: The terms between any two nonconsecutive terms of a geometric sequence.
Ex. 2, 6, 18, 54, 162
6, 18, 54 are the Geometric Mean between 2 and 162

11. Find two geometric means between –2 and 54
-2, ____, ____, 54 -2 54 4 NA r
The two geometric means are 6 and -18, since –2, 6, -18, 54
forms a geometric sequence

12. Geometric Series: An indicated sum of terms in a geometric sequence.
Example:
Geometric Sequence
3, 6, 12, 24, 48
VS Geometric Series

13. Recall
Vocabulary of Sequences (Universal)

14. # New people that receive joke Total # of people that received joke
Application: Suppose you a joke to three friends on Monday. Each of those friends sends the joke to three of their friends on Tuesday. Each person who receives the joke on Tuesday sends it to three more people on Wednesday, and so on.
Monday
Tuesday
# New people that receive joke
Day of Week
Total # of people that received joke
Monday
Tuesday
Wednesday 3 3 9
3 + 9 = 12 27
= 39

15. Find the sum of the first 10 terms of the geometric series 3 - 6 + 12 – 24+ …10 Sn -2

16. In the book Roots, author Alex Haley traced his family history back many generations to the time one of his ancestors was brought to America from Africa. If you could trace your family back 15 generations, starting with your parents, how many ancestors would there be? 2 15 Sn 2

17. a1 NA 8
39,360 3

18. 15,625 -5 ?? Sn

19. Recall the properties of exponents. When multiplying like bases add exponents

20. 15,625 -5 ?? Sn

21. UPPER LIMIT
(NUMBER)
SIGMA
(SUM OF TERMS)
NTH TERM
(SEQUENCE)
INDEX
LOWER LIMIT
(NUMBER)

22. If the sequence is geometric (has a common ratio) you can use the Sn formula
5+20 = 6
5+25 = 37 5 Sn 2

23. Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1
Geometric, r = ½

24. Infinite Series

25. 1, 4, 7, 10, 13, ….
Infinite Arithmetic
No Sum
3, 7, 11, …, 51
Finite Arithmetic
1, 2, 4, …, 64
Finite Geometric
1, 2, 4, 8, …
Infinite Geometric
r > 1
r < -1
No Sum
Infinite Geometric
-1 < r < 1

26. Find the sum, if possible:

27. Find the sum, if possible:

28. Find the sum, if possible:

29. Find the sum, if possible:

30. Find the sum, if possible: