1. What you really need to know!
A geometric sequence is a sequence in which the quotient of any two consecutive terms, called the common ratio, is the same. In the sequence 1, 4, 16, 64, 256, . . , the common ratio is 4.

2. Geometric Series

3. Geometric Sequence
The ratio of a term to it’s previous term is constant.
This means you multiply by the same number to get each term.
This number that you multiply by is called the common ratio (r).

4. Example: Decide whether each sequence is geometric.
4,-8,16,-32,…
-8/4=-2
16/-8=-2
-32/16=-2
Geometric (common ratio is -2)
3,9,-27,-81,243,…
9/3=3
-27/9=-3
-81/-27=3
243/-81=-3
Not geometric

5. Rule for a Geometric Sequence
un=u1r n-1
Example: Write a rule for the nth term of the sequence 5, 2, 0.8, 0.32,… . Then find u8.
First, find r.
r= 2/5 = 0.4
un=5(0.4)n-1
u8=5(0.4)8-1
u8=5(0.4)7
u8=5( )
u8=

6. One term of a geometric sequence is u4 = 3. The common ratio is r = 3
One term of a geometric sequence is u4 = 3. The common ratio is r = 3. Write a rule for the nth term. Then graph the sequence.
If u4=3, then when n=4, un=3.
Use un=u1rn-1
3=u1(3)4-1
3=u1(3)3
3=u1(27)
1/9=u1
un=u1rn-1
un=(1/9)(3)n-1
To graph, graph the points of the form (n,un).
Such as, (1,1/9), (2,1/3), (3,1), (4,3),…

7. Two terms of a geometric sequence are u2= -4 and u6= -1024
Two terms of a geometric sequence are u2= -4 and u6= Write a rule for the nth term.
Write 2 equations, one for each given term.
u2 = u1r2-1 OR -4 = u1r
u6 = u1r6-1 OR = u1r5
Use these equations & sub in to solve for u1 & r.
-4/r=u1
-1024=(-4/r)r5
-1024 = -4r4
256 = r4
4 = r & -4 = r
If r = 4, then u1 = -1.
un=(-1)(4)n-1
If r = -4, then u1 = 1.
un=(1)(-4)n-1
un=(-4)n-1
Both Work!

8. Please find the 15th term
5, 10, 20, 40
So, geometric sequence with u1 = 5 r = 2

9. Homework

10. Compound Interest

11. Compound Interest - Future Value Interest Interest Interest Interest
100
110
Interest
Interest
121
Interest
133.1
Amount $1000 1 2 3 4
1331
100
110
121
1210
100
110
1100
100
1000
Compounding Period
Compounding Period
Compounding Period
Compounding Period
Time(Years)

12. COMPOUND INTEREST FORMULA
FV is the Future Value in t years
P is the Present Value amount started with r is the annual interest rate
n number of times it compounds per year.

13. EXAMPLE Find the amount that results from the investment:
$50 invested at 6% compounded monthly after a period of 3 years.
$59.83

14. COMPARING COMPOUNDING PERIODS
Investing $1,000 at a rate of 10% compounded annually, quarterly, monthly, and daily will yield the following amounts after 1 year:
FV = PV(1 + r) = 1,000(1 + .1) = $

15. COMPARING COMPOUNDING PERIODS
Investing $1,000 at a rate of 10% compounded annually, quarterly, monthly, and daily will yield the following amounts after 1 year:

16. Interest Earned
Investing $1,000 at a rate of 10% compounded annually, quarterly, monthly, and daily will yield the following amounts after 1 year:

17. Homework

18. Sum of a Finite Geometric Series
The sum of the first n terms of a geometric series is
Notice – no last term needed!!!!

19. Formula for the Sum of a Finite Geometric Series
What is n? What is a1? What is r?
n = # of terms
a1 = 1st term
r = common ratio

20. Example
Find the sum of the 1st 10 terms of the geometric sequence: 2 ,-6, 18, -54
What is n? What is a1? What is r?
That’s It!

21. Example: Consider the geometric series 4+2+1+½+… .
Find the sum of the first 10 terms.

22. Geometric Series

23. Infinite Geometric Series
Consider the infinite geometric sequence
What happens to each term in the series?
They get smaller and smaller, but how small does a term actually get?
Each term approaches 0

24. What is happening to the sum?
Partial Sums
Look at the sequence of partial sums:
It is approaching 1 1
It’s CONVERGING TO 1.
What is happening to the sum?

25. Sum of an Infinite Geometric Series
Here’s the Rule
Sum of an Infinite Geometric Series
If |r| < 1, the infinite geometric series
a1 + a1r + a1r2 + … + a1rn + …
converges to the sum
If |r| > 1, then the series diverges (does not have a sum)

26. Converging – Has a Sum
So, if -1 < r < 1, then the series will converge. Look at the series given by
Since r = , we know that the sum is
The graph confirms:

27. Diverging – Has NO Sum
If r > 1, the series will diverge. Look at ….
Since r = 2, we know that the series grows without bound and has no sum.
The graph confirms:

28. Example Find the sum of the infinite geometric series 9 – 6 + 4 - …
We know: a1 = 9 and r = ?

29. You Try
Find the sum of the infinite geometric series 24 – – 3 + …
Since r = -½

30. Page 173 6G.2

31. Page 173 6G.2

32. Homework Page 173 All 2 - 8 REVIEW 6A (NO CALCULATOR)
REVIEW 6B (WITH CALCULATOR)