Sequences and Series - a New Type of Function
A sequence is a string of numbers in a specific order
Example 1: 1,3, 5, 7, 9, 11,…. Is an infinite sequence.
Example 2: 2,4,8,16,32,64,128
is an infinite sequence.
Example 3: 1,1,2,3,5,8,13,21,34,55,89,…. is an infinite sequence.
Generate two examples of sequences
1.
2.
Sequences are Discrete Functions
- We use 2 variables to describe a sequences n is the term number in the sequence (the independent
variable or x value) and tn the value of the nth term in the sequence (the dependent variable or y variable)
For Example 2 above the following table shows n and associated tn
n
1
2
3
4
5
6
7
8
tn
2
4
8
16
32
64
128
256
A sequence is a function or a relationship between term number (Position in the sequence) and term value.
Make function t tables for
n
n
the sequences in example 1
tn
tn
and example 3 above
There are many different types of sequences, but the two types that we will study are arithmetic sequences
and geometric sequences
Arithmetic Sequences
Example 1 above is an arithmetic sequence.
Example 4: another arithmetic sequence is 1.7, 2, 2.3, 2.6, 2.9, 3.2, ….
Example 4: another arithmetic sequence is 6, 2, -2, -6, -10, -14, -18,…..
Definition of an Arithmetic Sequence:
Create 2 more examples of arithmetic sequences
Finding the term value (tn) if you know the term number in an arithmetic sequence.
For example 1 above, find t27
For example 4 above find t36
For example 5 above find t60
Geometric Sequences
Example 2 above is a geometric sequence.
Example 6 144, 72, 36, 18, 9, 4.5, 2.25,….. is a geometric sequence
Example 7 100, -150, +225, -337.5, 506.25, ….. is a geometric sequence
Definition of a Geometric Sequence:
Create 2 more examples of geometric sequences
Finding the term value (tn) if you know the term number in a geometric sequence.
For example 2 above, find t11
For example 6 above find t20
For example 7 above find t15
Note: Example 3 above is a sequence that is neither geometric or arithmetic. It is called the Fibonacci
sequence. What is the rule between terms for a Fibonacci sequence.
Sequences - Adding Them Up = Series
Arithmetic sequence (Definition)
Example1: 7,10,13,16,19,22,….
Find the next term.
Find t42
Geometric Sequence (definition)
Example 2: 729, 243, 81, 27,……
Find the next term
Find t8
A series is the sum of a sequence (the number you get when you add up all the terms.) Since sequences are
infinite, we seldom add up all the terms. A Partial series (or partial sum of a sequence) Sn is the sum of the
first n terms in a sequence.
Sigma Notation for the partial sum of a sequence:
Write out the sigma notation for the 9th partial series of the sequence in example 1 above. What is the value of
this sum?
Formula for the partial series of an arithmetic sequence
Calculate S42 for the sequence in example 1
Write out the sigma notation for the seventh partial series of the sequence in example 2 above. What is the
value of this sum?
Formula for the partial series of a geometric sequence
Calculate S2- for the sequence in example 2
What do you think would happen to this sum as n gets very large (say n=100 or 1,000)
Convergent Geometric Series
When the ratio in a geometric sequence is greater than -1 and less than 1 the series (sum of all of the terms in
a sequence) converges, that is the sum adds to a finite number.
Formula for the sum of a convergent geometric series
For the geometric series in Example 2 above, find the sum of the series
For the geometric series with t1=90 and r=.1, find the sum of the series.