chapter 1: sequences and sets 1.1 sequences. sequences what number comes next? 1, 2, 3, 4, 5, ____...
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Chapter 1:Sequences and Sets
1.1 Sequences
Sequences
What number comes next?
1, 2, 3, 4, 5, ____
2, 6, 10, 14, 18, ____
1, 2, 4, 8, 16, ____
6
22
32
What comes next?
2, 5, 10, 17, 26, 37, ____
1, 2, 6, 24, 120, ____
2, 3, 5, 8, 12, ____
1, 1, 2, 3, 5, 8, 13, ____
50
720
17
21
The key to any sequence is to discover its pattern
• The pattern could be that each term is somehow related to previous terms
• The pattern could be described by its relationship to its position in the sequence (1st, 2nd, 3rd etc…)
• You might recognize the pattern as some well known sequence of integers (like the evens, or multiples of 10).
• You might be able to do all three of these ways!
2, 4, 6, 8, 10 …
• Can we define a given element in relation to previous elements.– The first element has to be 2 since it has no previous
elements– The second element is 2 more than the first element– The third element is 2 more than the second element. – In fact, each subsequent element is just two more than
the previous one.
2, 4, 6, 8, 10 …
• Can we represent this sequence in relation to its position?– At position 1 the value is 2– At position 2 the value is 4– At position 3 the value is 6– At position n the value is 2 * n
2, 4, 6, 8, 10 …
• Do we recognize this sequence as something familiar?
–Yes. It is the positive even numbers.
Mathematical Notation
• When we want to refer to terms in a sequence we usually use lower case letters (a, b, …) followed by a subscript indicating the position in the sequence we are referring to.
• a1 is the first term in a sequence
• a2 is the second term in a sequence
• an is the nth term in a sequence
2, 4, 6, 8, 10 …
• What is a1?
• What is a3?
• What is a5?
• What is an if n = 4?
• What is an-1 if n = 4?
2610
86
Recursive Formula
• A recursive formula for a sequence is one where each term is described in relation to a previous term (or terms)
2, 4, 6, 8, 10 …
Write each term in relation to its prior term (as a recursive formula)
• What is a1? a1=2 (no prior terms)
• What is a3? a3= a2+ 2
• What is a5? a5= a4+ 2
• What is an? an= an-1+ 2
Closed Formula
• A closed formula for a sequence is a formula where each term is described only by its relation to its position.
2, 4, 6, 8, 10 …
Write each term in relation to its position (as a closed formula)
• What is a1? a1=1* 2 (no prior terms)
• What is a3? a3= 3 * 2
• What is a5? a5= 5 * 2
• What is an? an= n * 2
Let’s try another
• Consider the sequence 5, 9, 13, 17…
• Write a recursive formula for this sequence.
a1=5
an= an-1 + 4
•Write a closed formula for this sequence
an= 4n + 1
And another
1, 3, 7, 15, 31, 63…
1 2 3 4 5 6 n
1 3 7 15 31 63 ?
Making a table can help to get a feel for what’s going on.
The recursive formula actually isn’t too bad.
• a1=1
• an= 2an-1 + 1
The closed formula is a little bit harder this time.
12 nna
Now try your hand at these.
Find both recursive and closed formulas for the following sequences.
2, 6, 10, 14, 18, ____
1, 2, 4, 8, 16, ____
1, 2, 6, 24, 120, ____
2, 6, 10, 14, 18, ____
Recursive Formula
Closed Formula
4
2
1
1
nn bb
b
24 nbn
Recursive Formula
Closed Formula
1
1
2
1
nn cc
c
)1(2 nnc
1, 2, 4, 8, 16, ____
Recursive Formula
Closed Formula
ndd
d
nn
1
1 1
!ndn
1, 2, 6, 24, 120, ____
Find a recursive formula from a closed formula
67 nanFirst write out the first several terms so you can understand the sequence
1 2 3 4 5
1 8 15 22 29
Find a recursive formula from a closed formula
67 nanEstablish the starting point in the pattern and then determine how each value relates the prior value
7
1
1
1
nn aa
a
Find a closed formula from a recursive formula
13
2
1
1
nn aa
a
1 2 3 4 5
2 7 22 67 202
These tend to be harder but you still want to establish and understand the sequence
Find a closed formula from a recursive formula
13
1
1
1
nn aa
a
At this point unless you are really clever we are kind of stuck, the pattern is not one you may recognize. Specialized methods are needed to complete these.
Common Mathematical Notion
• Summation: A summation is just the sum of the terms in a sequence.
• If the terms in the sequence are– 1, 2, 3, 4, 5, 6 then the summation is – 1+2+3+4+5+6 = 21
• If the terms in the sequence are– 1, 4, 9, 16, 25 then the summation is– 1+4+9+16+25 = 55
Summation is a very common Idea
• Because it is so common, mathematicians have developed a shorthand to represent summations (also called sigma notation)
n
i
i1
This is what the shorthand looks like, on the next few slides we will dissect it a bit.
Dissecting Sigma Notation
n
i
i1
The giant Sigma just means that this represents a summation
Dissecting Sigma Notation
n
i
i1
The i=1 at the bottom just states where is the sequence we want to start. If the value was 5 then we would start the sequence at the 5th position
Dissecting Sigma Notation
n
i
i1
The n at the top just says to what element in the sequence we want to get to. In this case we want to go up through the nth item.
Dissecting Sigma Notation
n
i
i1
The portion to the right of the sigma is the closed formula for the sequence you want to sum over.
n
i
i1
Dissecting Sigma Notation
So this states that we want to compute the closed closed formulaformula for each element from 11 to nn.
Dissecting Sigma Notation
n
i
i1
The portion to the right of the sigma is the closed closed formulaformula for the sequence you want to sum over.
Dissecting Sigma Notation
n
i
i1
Thus our summation is
1 + 2 + 3 + … + n
If I told you that n had the value of 5, then the summation would be
1 + 2 + 3 + 4 + 5 = 15
Let’s try a few. Compute the following summations
5
1
)2(i
i
7
1
2 )1(i
i
2576543
147503726171052
How would you write the following sums using sigma notation?
5+10+15+20+25+30+35+40
1+8+27+64+125+216
8
1
)5(i
i
6
1
3)(i
i
So why are sequences important
• Identifying patterns is an essential tool for anyone
• Developing a vocabulary to represent and analyze these sequences is the key to speaking the language of mathematics.