1. Warm up
1. Determine if the sequence is arithmetic. If it is, find the common difference.
35, 32, 29, 26, ...
2. Given the first term and the common difference of an arithmetic sequence find the first five terms and the explicit formula.
a1 = 28, d = 10
yes, d = -3
First Five Terms: 28, 38, 48, 58, 68, Explicit: an =28 + (n-1)10

2. Lesson 12-2 Geometric Sequences & Series
Objective: To find the nth term and the geometric means of a geometric sequence
To find the sum of n terms of a geometric series

3. Geometric Sequence
Geometric sequences increase by a constant factor called the common ratio (r)
an = ran-1
Geometric Sequences are also called geometric progressions.
Common ratio

4. Geometric Sequences and Series
A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a constant nonzero real number.
1, 2, 4, 8, 16 … is an example of a geometric sequence with first term 1 and each subsequent term is 2 times the term preceding it. The common ratio (r), is 2

5. Finding the Common Ratio
In a geometric sequence, the common ratio can be found by dividing any term by the term preceding it.
The geometric sequence 2, 8, 32, 128, …
has common ratio r = 4 since

6. Geometric Sequences and Series
nth Term of a Geometric Sequence
In the geometric sequence with first term a1 and common ratio r, the nth term an, is

7. Using the Formula for the nth Term
Example Find a5 and an for the geometric
sequence 4, –12, 36, –108 , …
Solution Here a1= 4 and r = 36/ –12 = – 3. Using
n=5 in the formula
In general

8. Modeling a Population of Fruit Flies
Example A population of fruit flies grows in such a
way that each generation is 1.5 times the previous
generation. There were 100 insects in the first
generation. How many are in the fourth generation.
Solution The populations form a geometric sequence
with a1= 100 and r = Using n=4 in the formula
for an gives
or about 338 insects in the fourth generation.

9. Geometric Means
In a geometric sequence the terms between two nonconsecutive terms are called geometric means.

10. Practice
Write a sequence that has two geometric means between 128 and 54.
Find the common ratio.
Use r to find the other terms.
128, 96,72,54

11. Geometric Series
A geometric series is the sum of the terms of a geometric sequence .
In the fruit fly population model with a1 = 100 and r = 1.5, the total population after four generations is a geometric series:

12. Geometric Sequences and Series
Sum of the First n Terms of an Geometric Sequence
If a geometric sequence has first term a1 and common ratio r, then the sum of the first n terms is given by or
where

13. Finding the Sum of the First n Terms
Example Find the sum of the first six terms of a geometric series if a1 = 6 and r=3.
Solution

14. Geometric vs. arithmetic sequences
The difference is in how they grow
Arithmetic sequences increase by a constant amount
an = 3n
The sequence {an} is { 3, 6, 9, 12, … }
Each number is 3 more than the last
Of the form: f(x) = dx + a
Geometric sequences increase by a constant factor
bn = 2n
The sequence {bn} is { 2, 4, 8, 16, 32, … }
Each number is twice the previous
Of the form: f(x) = arx

15. Practice
Determine if the sequence is geometric. If it is, find the common ratio.
1) −1, 6, −36, 216, ) −1, 1, 4, 8, ...
Given the first term and the common ratio of a geometric sequence find the first five terms and the explicit formula.
3)a1= 0.8, r = − ) a1= 1, r = 2
R = -6
Not geometric
First Five Terms: 0.8, −4, 20, −100, Explicit:an=0.8 ×(−5)^n− 1
First Five Terms: 1, 2, 4, 8, 16 Explicit: a n =2^(n− 1)