1. Sect.R10 Geometric Sequences and Series

2. Definition of Sequence
An infinite sequence is a function whose domain is the set of positive integers.
a1, a2, a3, a4, , an, . . .
terms
The first three terms of the sequence an = 2n2 are
a1 = 2(1)2 = 2
a2 = 2(2)2 = 8
finite sequence
a3 = 2(3)2 = 18.
Definition of Sequence

3. Definition of Geometric Sequence
A sequence is geometric if the ratios of consecutive terms are the same.
2, 8, 32, 128, 512, . . .
geometric sequence
The common ratio, r, is 4.
Definition of Geometric Sequence

4. The nth Term of a Geometric Sequence
The nth term of a geometric sequence has the form
an = a1rn - 1
where r is the common ratio of consecutive terms of the sequence.
a1 = 15
15, 75, 375, , . . .
a2 = 15(5)
a3 = 15(52)
a4 = 15(53)
The nth term is 15(5n-1).
The nth Term of a Geometric Sequence

5. Example: Finding the nth Term
Example 1: Find the 9th term of the geometric sequence
7, 21, 63, . . .
a1 = 7
an = a1rn – 1 = 7(3)n – 1
a9 = 7(3)9 – 1 = 7(3)8
= 7(6561) = 45,927
The 9th term is 45,927.
Example: Finding the nth Term

6. Example: Finding the nth Term
Example 2: Find the 30th term of the geometric sequence
4, 16, 64, . . .
Example: Finding the nth Term

7. Definition of Summation Notation
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

8. The Sum of a Finite Geometric Sequence
The sum of a finite geometric sequence is given by
= ?
n = 8
a1 = 5
The Sum of a Finite Geometric Sequence

9. Definition of Geometric Series
The sum of the terms of an infinite geometric sequence is called a geometric series.
If |r| < 1, then the infinite geometric series
a1 + a1r + a1r2 + a1r a1rn
has the sum
Definition of Geometric Series

10. Example: Sum of Infinite Geometric Series
Example 3: Find the sum of
Example: Sum of Infinite Geometric Series

11. Example: Sum of Infinite Geometric Series
Example 4: Find the sum of
Example: Sum of Infinite Geometric Series

12. Graphing Utility: Terms and Sum of a Sequence
Graphing Utility: Find the first 5 terms of the geometric sequence an = 2(1.3)n.
end value
variable
beginning value
List Menu:
Graphing Utility: Find the sum
upper limit
List Menu:
variable
lower limit
Graphing Utility: Terms and Sum of a Sequence

13. Example: Sum of Infinite Geometric Series
Example 5: Find the sum using a graphing utility
Example: Sum of Infinite Geometric Series

14. Example: Sum of Infinite Geometric Series
Example 6: Given the explicit formula for the geometric sequence, find the first 5 terms and the 10th term.
Example: Sum of Infinite Geometric Series

15. Example: Sum of Infinite Geometric Series
Example 7: Given the recursive formula for the geometric sequence, find the first 5 terms and the explicit formula
Example: Sum of Infinite Geometric Series

16. Assignment: Worksheet 10-R
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