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8.1: Sequences
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2008
Craters of the Moon National Park, Idaho
A sequence is a list of numbers written in an explicit order.
1 2 3,, , ... , , ... n na a a a a
nth term
Any real-valued function with domain a subset of the positive integers is a sequence.
If the domain is finite, then the sequence is a finite sequence.
In calculus, we will mostly be concerned with infinite sequences.
A sequence is defined explicitly if there is a formula that allows you to find individual terms independently.
2
1
1
n
na n
Example:
To find the 100th term, plug 100 in for n:
100
100 2
1
100 1a
1
10001
A sequence is defined recursively if there is a formula that relates an to previous terms.
We find each term by looking at the term or terms before it:
1 2 for all 2n nb b n Example: 1 4b
1 4b
2 1 2 6b b
3 2 2 8b b
4 3 2 10b b
You have to keep going this way until you get the term you need.
An arithmetic sequence has a common difference between terms.
Arithmetic sequences can be defined recursively:
3d Example: 5, 2, 1, 4, 7, ...
1n na a d
ln 6 ln 2d ln 2, ln 6, ln18, ln 54, ...6
ln2
ln 3
or explicitly: 1 1na a d n
An geometric sequence has a common ratio between terms.
Geometric sequences can be defined recursively:
2r Example: 1, 2, 4, 8, 16, ...
1n na a d
1
2
10
10r
2 110 , 10 , 1, 10, ... 10
or explicitly:1
1n
na a d
Example: If the second term of a geometric sequence is 6 and the fifth term is -48, find an explicit rule for the nth term.
41
1
48
6
a r
a r
3 8r
2r
2 12 1a a r
16 2a
13 a
13 2
n
na
You can determine if a sequence converges by finding the limit as n approaches infinity.
Does converge?2 1
n
na
n
2 1limn
n
n
2 1limn
n
n n
2 1lim limn n
n
n n
2 0
2
The sequence converges and its limit is 2.
Absolute Value Theorem for Sequences
If the absolute values of the terms of a sequence converge to zero, then the sequence converges to zero.
Don’t forget to change back to function mode when you are done plotting sequences.
