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. Geometric Sequence
A sequence is geometric if the ratios of consecutive terms are the same.
The number r is the common ratio.

4. 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

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

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

7. 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 =

8. 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

9. Ex. 5
The fourth term of a geometric sequence is 125, and the 10th term is 125/64. Find the 14th term. (Assume all terms are positive)
a4 = 125
a10 = a4r6
a14 = a10r4
a10 =
a14 =

10. sum of a finite geometric series

11. 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

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

13. 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

14. Find the sum, if possible:

15. Find the sum, if possible:

16. Find the sum, if possible:

17. Find the sum, if possible:

18. 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

19. 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

21. 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

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