1. Geometric Sequences

2. A Geometric Sequence is
a sequence of numbers in which each term is formed by multiplying the previous term by the same number or expression.
The consecutive terms have a common ratio.
1, 3, 9, 27, 81, 243, ...
The terms have a common ratio of 3.

3. Geometric Sequence Example Is the following sequence geometric?
4, 6, 9, 13.5, 20.25, …
Yes, the common ratio is 1.5

4. Geometric Sequence
To find any term in a geometric sequence, use the formula
an = a1 rn–1
where r is the common ratio.

5. Example
Find the twelfth term of the geometric sequence whose first term is 9 and whose common ratio is 1.2.
an = a1 rn–1
a1 = 9 r = 1.2
a9 = 9 • 1.211
a12 = 66.87
To find the sum of a geometric series, we can use summation notation.

6. Example
Which can be simplified to:

7. Evaluate the sum of:
Convert this to =

8. Series

9. Series

10. Series
Definition: A series is a partial sum of the first n terms of a sequence.
General term: nth partial sum: Sn =
nth partial sum of arithmetic sequence:
Example: nth partial sum of an = n. Sn =

11. Series nth partial sum of geometric sequence:
Sum of an infinite geometric sequence:
If |r|<1,
If |r| 1, a geometric series has no infinite sum.
Example:

12. Series
Product notation: