Maggie Johnson
Handout #28
CS103A
Sequences and Summations
Key topics:
•
Sequences
•
•
Arithmetic and Geometric Progressions
•
Fractals
A mathematician, like a poet or a painter, is a maker of patterns.
G.H. Hardy
A Mathematician’s Apology (1940)
Sequences
Imagine a person (with a lot of time on their hands) that decides to count their ancestors.
She has
two parents, four grandparents, eight greatgrandparents, and so forth.
We could write these
numbers in a row: 2, 4, 8, 16, 32, 64, …
(where the … means and so forth).
To express a pattern of numbers in this manner, we often label the position of each number in the
row as in the following table.
1
2
3
4
5
6
2
4
8
16
32
64
When represented in this manner it is easy to recognize a formula that would give us the kth
element in the row: A
k
= 2
k
.
Note that we are just making an observation based on evidence in
guessing this formula.
We would need to do a proof to be absolutely certain.
A sequence
is an ordered list of elements written in a row, such that each element has a unique
position in the list.
We use a
k
to denote a single element of a sequence called a term
.
The k in
a
k
is called a subscript
or index
.
An explicit formula
for a sequence is a rule that shows how
the value of a
k
is derived from k.
What is the programming analogy of a sequence?
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View Full DocumentA common problem in computer science is determining an explicit formula given only the first
few elements of a sequence.
When trying to find such a formula we try to find a pattern.
A good
place to start is in asking the following questions:
•
Are there runs of the same values?
•
Are terms obtained from previous terms by adding the same amount, or an amount that
depends on position in the sequence?
•
Are terms obtained from previous terms by multiplying by a particular amount?
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 Fall '07
 Plummer,R
 Computer Science, Geometric progression, Fractal, Koch snowflake, helge von koch

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