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Introduction to Recursion
Introduction to Recursion
In order to understand recursion, we’ll need to
take a brief look at sequences.
A sequence is a set of numbers that
have a specific order.
Introduction to Recursion
A very simple example of a sequence is:
{1, 2, 3, 4, 5, …}
This sequence happens to be infinite in
size, but that is not true of all sequences.
Introduction to Recursion
It’s very important to understand the notation
used in sequences.
Perhaps most importantly, you’ll need to
know the difference between a term and
a term number.
Introduction to Recursion
Consider another simple example:
{4, 5, 6, 7, 8,…}
The terms are the numbers themselves.
The term number indicates the position
of a term in the sequence.
Introduction to Recursion
Consider another simple example:
{4, 5, 6, 7, 8,…}
So the first term is 4.
The second term is 5, etc.
Introduction to Recursion
Consider another simple example:
{4, 5, 6, 7, 8,…}
A convention is to use a capital letter to
represent the term, such as T.
A subscript indicates the term number.
Introduction to Recursion
Consider another simple example:
{4, 5, 6, 7, 8,…}
In our case,
T1 = 4, T2 = 5, T3 = 6, etc.
Introduction to Recursion
Consider another simple example:
{4, 5, 6, 7, 8,…}
A recursion uses the same process to
work from one term to the next in a
sequence.
In our example, the process is to add 1.
Introduction to Recursion
Consider another simple example:
{4, 5, 6, 7, 8,…}
4 + 1 = 5,
5 + 1 = 6.
6 + 1 = 7,
etc.
Introduction to Recursion
Consider another simple example:
{4, 5, 6, 7, 8,…}
We want a notation to show this
relationship. We use this notation to
show that the value of one term depends
on the value of the previous term.
Introduction to Recursion
Consider another simple example:
{4, 5, 6, 7, 8,…}
For example, you need to know the first
term in order to determine the second
term. You need to know the second term
to determine the third term, etc.
Introduction to Recursion
Consider another simple example:
{4, 5, 6, 7, 8,…}
Using our notation to show the
relationship between a term and its
preceding term:
T2 = T1 + 1
T3 = T2 + 1
etc.
Introduction to Recursion
Consider another simple example:
{4, 5, 6, 7, 8,…}
We want a generic form to show this
“add 1” relationship for the sequence.
In general, the term in the nth position,
Tn, depends on the term in the previous
position, Tn-1.
Introduction to Recursion
Consider another simple example:
{4, 5, 6, 7, 8,…}
In short,
Tn = Tn-1 + 1
Introduction to Recursion
Consider another simple example:
{4, 5, 6, 7, 8,…}
In short,
Tn = Tn-1 + 1
If you want to determine the 10th term,
for example, take the 9th term and add 1.
Introduction to Recursion
Consider another simple example:
{4, 5, 6, 7, 8,…}
Notice that if we know this recursive
rule and the starting point of the
sequence (the first term), we can
determine the entire sequence.
Introduction to Recursion
Example 2
{2, 5, 8, 11,…}
What is the recursive rule?
Introduction to Recursion
Example 2
{2, 5, 8, 11,…}
We get from one term to the next by
adding 3, so the rule is
Tn = Tn-1 + 3
Introduction to Recursion
Example 3
{5, 1, -3, -7,…}
We get from one term to the next by
subtracting 4, so the rule is
Tn = Tn-1 - 4
Introduction to Recursion
Example 4
{2, 4, 8, 16,…}
In this case we get from one term to the
next by multiplying by 2, so the rule is
Tn = 2Tn-1
Introduction to Recursion
Example 4
{8, 4, 2, 1,…}
In this case we get from one term to the
next by dividing by 2. This is the same
as multiplying by 1/2 so the rule is
Tn = (1/2)Tn-1
Introduction to Recursion
Example 4
{2, 4, 16, 256,…}
In this case we get from one term to the
next by squaring the previous term so
the rule is
Tn = (Tn-1)2