1. How do I find the sum & terms of geometric sequences and series?
Essential Question: What is a sequence and how do I find its terms and sums?
How do I find the sum & terms of geometric sequences and series?

2. Geometric Sequences
Geometric Sequence– a sequence whose consecutive terms have a common ratio.

3. Ex. 1 2, 4, 8, 16, …, formula?, … 12, 36, 108, 324, …, formula?, …
Are these geometric?
2, 4, 8, 16, …, formula?, …
Yes 2n
Yes 4(3)n
12, 36, 108, 324, …, formula?, …
No (-1)n /3
1, 4, 9, 16, …, formula? , …
No n2

4. Finding the nth term of a Geometric Sequence
an = a1rn – 1

5. Ex. 2b
Write the first five terms of the geometric sequence whose first term is a1 = 9 and r = (1/3).

6. Ex. 3
Find the 15th term of the geometric sequence whose first term is 20 and whose common ratio is 1.05
an = a1rn – 1
a15 = (20)(1.05)15 – 1
a15 =

7. Ex. 4 Find a formula for the nth term. 5, 15, 45, … an = a1rn – 1
an = 5(3)n – 1
What is the 9th term?
an = 5(3)n – 1
a9 = 5(3)8
a9 = 32805

8. Finite vs. Infinite Geometric Series
What was the difference between an infinite sequence and a finite sequence?
A finite sequence has a specific number of terms whereas an infinite sequence listing terms continually
Finite and infinite series follow the same pattern

9. sum of a finite geometric series

10. Ex. 6
Find the sum of the first 12 terms of the series 4(0.3)n
= 4(0.3)1 + 4(0.3)2 + 4(0.3)3 + … + 4(0.3)12
= 1.714

11. Ex. 7
Find the sum of the first 5 terms of the series 5/ …
r = 5/(5/3) = 3
= 605/3

12. Ex 8.
… n= 5
3. a1= -2, r = 3, Sn= -242

14. r=3

15. What would happen if the series was Infinite? Could you find the sum?
Arithmetic Infinite: No Sum
Geometric Infinite: There can be a sum depending on r value.
Geometric – If r > 1, then there is no sum (called Divergent)
Geometric – If r< 1 then there is a sum (called Convergent)

16. Formula for Infinite Convergent Geometric Series:

17. Convergent 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 - Divergent
Convergent
Infinite Geometric
-1 < r < 1

18. Decide if it is convergent or divergent, then Find the sum, if possible:

19. Decide if it is convergent or divergent, then Find the sum, if possible:

20. Decide if it is convergent or divergent, then Find the sum, if possible:

21. Decide if it is convergent or divergent, then Find the sum, if possible:

22. … 1.
4. Find the common ratio for the infinite geometric series if a1=81, S=60.75 3.

23. The Bouncing Ball Problem – Version A
A ball is dropped from a height of 50 feet. It rebounds 4/5 of
it’s height, and continues this pattern until it stops. How far
does the ball travel? 50 40 40 32 32
32/5
32/5

24. The Bouncing Ball Problem – Version B
A ball is thrown 100 feet into the air. It rebounds 3/4 of
it’s height, and continues this pattern until it stops. How far
does the ball travel?
100
100 75 75
225/4
225/4

25. upper limit of summation lower limit of summation
The sum of the first n terms of a sequence is represented by summation notation.
upper limit of summation
lower limit of summation
index of summation
Definition of Summation Notation

26. Write out a few terms.
If the index began at i = 0, you would have to adjust your formula