Sequences
section 9.4
Sequences are just patterns of numbers. Look at the examples below:
1) 5, 10, 15, 20, 25
k
2) 2, 4, 8, 16, 32, …, 2 , …
1

: k  1,2,3,...
k

3) 
4)
a a , a ,...a ,...
1, 2
3
k
Important things to recognize:
a) Finite vs. Infinite sequences
b) Differentiating between the number of the term and the value of the term
c) Writing “rules” for the kth term in a sequence
Defining a Sequence Explicitly
This allows you to use a formula to determine any term in
the sequence, simply by plugging in the number of the
term.
Example: Find the first 6 terms and the 100th term of the sequence, ak  , in which ak  k  1 .
2
Defining a Sequence Recursively
This allows you to find the next term in the sequence by
using the preceding term.
Example: Using the first sequence in the notes: 5, 10, 15, 20, 25
Example 2: Find the first 6 terms and the 100th term of the sequence defined recursively by:
b1  3
bn  bn1  2, for all n  1
Special Types of Sequences: Arithmetic and Geometric
Arithmetic:
A sequence where pairs of successive terms have a _______________
_________________.
a, a  d, a  2d, a  3d,...a  (n  1)d,...
All arithmetic sequences can be defined explicitly and recursively:
Explicit Rule:
Recursive Rule:
Example: For the sequence: -6, -2, 2, 6, 10, …
Find the following:
a) The common difference
b) The 10th term
c) A recursive rule for the nth term
d) An explicit rule for the nth term
Geometric:
A sequence where pairs of successive terms have a ________________
__________________.
a, a r, a r , a r ,...a r
2
3
n 1

,...
All geometric sequences can be defined explicitly and recursively:
Explicit Rule:
Recursive Rule:
Example: For the sequence: 3, 6, 12, 24, 48, … Find the following:
a) The common ratio
b) The 10th term
c) A recursive rule for the nth term
d) An explicit rule for the nth term