Transcript Introduction to Algebra
Sequences Ordered Patterns
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The art of asking the right questions in mathematics is more important than the art of solving them −
Georg Cantor (1845-1918)
Sequences 2
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Sequences
What is a sequence?
A pattern of objects arranged in an ordering corresponding to the ordering of the natural numbers
Definitions: An infinite sequence is a function whose domain is the set of natural numbers A finite sequence is a function with domain D = { 1, 2, 3, ..., n } for some positive integer n
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Sequences
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Examples ABCD 1 3 5 7 9 11 13 ... 15 17 ... repeated group ... odd natural numbers ... prime numbers ... multiples of 9 ... Fibonacci sequence
What are the next four characters in each of the above sequences?
Identifying the pattern in the sequence allows for prediction of later values
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Sequences As Functions
Terminology and Notation
Applications of sequences generally do not require graphing, so the x-y notation is dropped
Since the domain is the set of natural numbers , each domain element is an integer n
Functional value is then written f(n) =
a
n
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Sequences As Functions
Terminology and Notation
Range elements
a
n are called terms
Terms are any kind of objects
Examples: mile markers, fence posts, customers, integers, database records
Terms can be arranged in “sequential” order via the subscript n – a sort of number tag Common notation:
{
a
n k
}
n=1
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Sequences As Functions
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As a Function
Relates each member n of the domain with exactly one term
a
n in the range Domain = N Range 2 3 1
a
1
a
2
a
3 S =
{
a
1 ,
a
2 ,
a
3 , …
}
Questions: Is S a relation ?
YES Is S a function ?
YES
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Sequences As Functions
Example
Consider the following sequence -1 0 1 2 3 4 5 6 7 8 9 n
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-50 -40
a
3 (-37.5)
a
7 (-24)
a
1 (-14)
a
2 (0)
a
5 (5)
a
6 (12)
a
8 (26)
a
4 (38)
a
9 (44) x
Sequence is random – no recursion !
{
9
a
n
}
n=1 = -14, 0, -37.5, 38, 5, 12, -24, 26, 44
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Recursive Sequences
What is recursion ?
Application of a repetitive pattern for generating successive terms
Patterns generate a new term based on
the value of its predecessors and a rule Additive – arithmetic sequences Multiplicative – geometric sequences
Generalized Fibonacci & other patterns
Not all sequences are recursive, e.g. random sequences with no general term
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Sequences As Patterns
Finding the General Term
Suppose we start with the sequence
a
1 = 2 ,
a
2 = 4 ,
a
3 = 8 ,
a
4 = 16
What is the general term
a
n for any positive integer n ?
Note that we can write the sequence as:
a
1 = 2 1 ,
a
2 = 2 2 ,
a
3 = 2 3 ,
a
4 = 2 4 , …
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Sequences As Patterns
Finding the General Term
Note that we can write the sequence as:
a
1 = 2 1 ,
a
2 = 2 2 ,
a
3 = 2 3 ,
a
4 = 2 4 , …
So we should have:
a
n general term = 2 n as the
If f is the sequence function, then f(n) =
a
n = 2 n is the general term of the sequence
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Finding the General Term
Example 1
Given:
a
1 = 2 ,
a
2 = 5 ,
a
3 = 8 ,
a
4 = 11
Find the n th term
a
n
Note that
a a
2
a
3 4 – –
a
1
a
2 –
a
3 = 5 – 2 = 3 = 8 – 5 = 3 = 11 – 8 = 3
Thus
a
n
+
1 –
a
n = 3 for n ≥ 1 that is, the common difference is 3
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Finding the General Term
Example 1
So
a
n
+
1 –
a
n = 3
a a a a a
2 3 4 5 n for n ≥ 1
a
n
+
1 =
a
n
+
3 = =
a
1
a
2 = =
a
3
a
3 =
a
1
+ +
3 = 3 =
a
1
a
1
+ +
1 (3) = 5 2 (3) = 8
+
3 =
a
1
+
3 (3) = 11
+
3 =
a
1
+
• • • • • • • • • • 4 (3) = 14
+
( n – 1 )3 = 2
+
( n – 1 )3
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Finding the General Term
Example 1 Question:
a
n = 2
+
( n – 1 )3 Does this work for
a
1 ,
a
2 ,
a
3 ?
Note: The general inductively form of
a
n was found from specific values Question: What is
a
21
a
21 = 2
+
?
( 21 – 1 )3 Note: = 62 We find the specific deductively from the value of
a
21 general form
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Finding Sequence Terms
From the General to the Specific
Example 2
Find the first four terms for:
a
n = 3(n – 1)
+
5
a
1
a
2
a
3
a
4 = 3( 1 – 1)
+
= 3( 2 – 1)
+
= 3( 4 – 1)
+
5 = 5 5 = 8 = 3( 3 – 1)
+
5 = 11 5 = 14 Question: What is
a
21 ?
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Finding Sequence Terms
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From the General to the Specific
Example 3
Find the first four terms for: ( –1) n
a a a a
1 2 3 4 = = = = ( ( ( ( –1) –1) –1) –1) 1 2 3 4 1 1 1 2 1 3 1 4 = –1 = = = 1 2 – 1 4 1 3 1 n Question: What is
a
21 ?
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Recursive Sequences
Finding Terms from Preceding Terms
Consider the sequence 1 1 2 3 5 8 13 21 34 ...
Here we have
a
1 = 1 ,
a
2 = 1 ,
a
3 = 2 ,
a
4 = 3 ,
a
5 = 5 , ...
So, how are these related? Well ... note that
a
3 =
a
2
+
a
1 ,
a
4 =
a
3
+
a
2 ,
a
5 =
a
4
+
a
3 , ...
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Recursive Sequences
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Finding Terms from Preceding Terms
a
3 =
a
2
+
a
1 ,
a
4 =
a
3
+
a
2 ,
a
5 =
a
4
+
a
3 , ...
Generally appears that, starting with n = 3 ,
a
n =
a
n –1
+
a
n –2
Functionally, it appears that f(n) = f(n – 1)
+
f(n – 2) for n ≥ 3
This is a recursive function ... in this case the basic Fibonacci Sequence
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Recursive Sequences
Examples 1.
a
1 = –1 and
a
n =
a
n –1
+
4 Find the first four terms
a
1
a
2
a
3 = –1 , =
a
1 =
a
2
+ +
4 = 3 , 4 = 7 ,
a
4 =
a
3
+
4 = 11 f(n) =
a
n ● ● ● ●
Notice anything about the graph ?
Sequences
n
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Recursive Sequences
Examples 1.
a
1 = –1 and
a
n =
a
n –1
+
4 What is ∆f from ∆f ∆n =
a
n
+
1 (n
+
–
a
n 1) – n n to n
+
= 4 1 = 4 1 ?
f(n) =
a
n ● ● ● If we were to allow n = 0 , what would f(0) be ?
● n
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Recursive Sequences
Examples 2.
a
1
a
n = 0 ,
a
2 = 2
a
n –1 = 1
+
a
n and –2 f(n) =
a
n Find the first five terms
a a
3 1 = 0
,
= 2
a
2
a
2
+
= 1 ,
a
1 = 2 , … ●● ● ●
Is f(n) =
a
4
a
n = 2
a
3
+
a linear function ?
a
2 = 5 ,
a
5 = 2
a
4
+
a
3 = 12 ● n
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Recursive Sequences
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Examples 3.
a
n = 3
a
n –1 and
a
1 = 1 27 Find the first five terms
a
1 = 1 27
, a
2 = 3
a
1 = 3 27 =
a a a a
3 4 5 n 1 f(n) =
a
n ● = 3
a
2 = 3(3
a
1 ) = 3 2
a
1 = 9 1 3 ● = 3
a
3 = 3(3 2 = 3
a
4 = 3 n –1 = 3 4
a
1
a
1
a
1 ) = 3 3
a
1 = 3 = 1 Question: ●● ● Is f(n) =
a
n a linear function ?
n
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Arithmetic Sequences
Definition
An arithmetic sequence is a function defined on the set of positive integers of form f(n) =
a
n =
a
n –1
+
d where d is the common difference
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Arithmetic Sequences
Arithmetic Sequence:
a
n =
a
n –1
+
d
Clearly
a
n –
a
n –1 = d for n ≥ 2
F n = 1 ,
a
1 is given independently
F n > 1 ,
a
n is computed recursively for each successive n
B induction ,
a
n for all n > 1 =
a
1
+
(n – 1)d
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Arithmetic Sequences
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Arithmetic Sequence:
a
n =
a
n –1
+
d
Example:
a
2 =
a
1
+
f(1) =
a
1 3 = – 4
+
= –4 3 = –1 , d = 3
a
3 =
a
2
+
3 = (
a
1
+
3)
+
3 =
a
1
+
2 (3)
a a a
4 5 n = =
a
4
+
=
a
3
a
1
+ +
= 2 3 = ( (
a
1
+
2(3) )
+
3 =
a
1 3 = ( (
a
1
+
3(3) • • • • • • • • • • ( n – 1 )3 = –4
+
)
+
3 =
a
1 3( n – 1 )
+
3 (3)
+
= 5 4 (3) = 8 Question: Is f(n) linear ?
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Arithmetic Sequences
Arithmetic Sequences as Functions
Given arithmetic sequence
{
a
n
}
where
a
n =
a
n –1
+
d =
a
1
+
(n – 1)d Function f(n) is f(n) =
a
n =
a
1
+
(n – 1)d f(n) can be written as where f(k) =
a
1 k = n – 1
+
kd = dk
+
a
1
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Arithmetic Sequences
Arithmetic Sequences as Functions where f(k) =
a
1 k = n – 1
+
kd = dk
+
a
1 The rate of change of f(k) is d and f(0) =
a
1 Thus f(k) = dk
+
a
1 a linear function with slope d and vertical intercept (0,
a
1 )
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Arithmetic Sequences
Example Given arithmetic sequence
{
a
n
}
with
a
1 = 5 and d = 3 , map the sequence function f(n) = dk
+
a
1 f(k) f(k) = 3k
+
5
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k k 1 2 3 4 5 6 7 8 9 10 ….
0 1 2 3 4 5 6 7 8 9 ….
5 8 11 14 17 20 23 26 29 32 ….
Sequences
k
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Geometric Sequences
Definition
A geometric sequence is a function defined on the set of positive integers of form f(n) =
a
n = r
a
n –1 where r is the common ratio and
a
1 = c is a constant
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Geometric Sequences
A geometric sequence is a function defined on the set of positive integers of form f(n) =
a
n = r
a
n –1
By induction r
a
we can show that n –1 = r n –1
a
1 Note:
a
n
a
n –1 = r , the ratio of successive terms
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Geometric Sequences
Example f(1) =
a
1
a
3 =
a
2 = = 1 , r = 3 9 3
a
1 3
a
2 = 1 = 3 = 9 3
(
3
a
1
)
1 3 = 3 2
a
1 = 3 1 3 By induction
a
n = 3 n –1
a
1 = 1 9 3 n –1 = 1 Question: Is f(n) =
a
n a linear function ?
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Geometric Sequences
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Geometric Sequences as Functions
Given geometric sequence
{
a
n
}
where
a
n = r
a
n –1 = r n –1
a
1 The function f(n) =
a
n = r n –1
a
1 can be written, with k = n – 1 , as making f(k) f(k) = r k
a
1 =
a
1 r k an exponential function The rate of change of f(k) is r k (r – 1)
a
1 and so is never constant
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n k f(k)
Geometric Sequences
f(k)
Example Given geometric sequence
{
a
n
}
with
a
1 = 3 and r = 2 Sequence function: f(n) =
a
1 (r n –1 ) or f(k) =
a
1 (r for k = n – 1 k ) 1 2 3 4 5 6 7 ….
0 1 2 3 4 5 3 6 12 24 48 96 6 ….
192 ….
400 200
5
k ) k
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Retrospective
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Sequences in Review
Arithmetic Sequence
Successive terms with common difference
Sequence function is linear
Geometric Sequence Successive terms with common ratio
Sequence function is exponential
Other Sequences
Many recursion patterns possible Random sequences without pattern
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Think about it !
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