Task 5 presentation slides
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Monday 25 March 2013
Sunday 17 March 2013
SEQUENCES
Sequences
A sequence is a method of representing a list of ordered terms by using the
pattern of a1, a2, a3, …… such as 1, 2, 3, 4,
5, …… , 3, 5, 7, 9, 11 and so on.
Example 1:
Given xn
= n, show the sequence of {xn} from n = 1.
Solution:
When we
come across with sequence such as x1, x2, x3,
…. We often denote a sequence with {xn},
which means that a sequence is actually
the list of elements in a set.
To show the
sequence of xn, just substitute numbers starting from 1 to n. Therefore we can get a sequence
like this:
{ xn
} = x1, x2, x3, x4, x5,
…
{ xn
} = ( x1 = 1 ), ( x2 = 2 ), ( x3 = 3 ), ( x4
= 4 ), ( x5 = 5 ), …
{ xn
} = 1, 2, 3, 4, 5, …
Example 2:
an =
2x – 1, find {an} starting from n = 2.
Solution:
Just as the
example above, substitute the values starting from n = 2. Therefore you would
get:
{an}
= 2(2)-1, 2(3)-1, 2(4)-1, 2(5)-1, 2(6)-1, …
{an}
= 3, 5, 7, 9, 11, …
Types of Sequence
There are
commonly 3 types of sequences. The geometric sequence, the arithmetic sequence
and the special number sequence.
Geometric Progression
Geometric progression is a sequence with a pattern like
a, arn, arn+1, arn+2, ……, where a is the
intial term of the sequence and r is the common ratio. Each of the term has a
common ratio with the terms just before and after them.
Example 3:
Let the
initial term a = 3 and the common ratio is 2. Write a sequence of geometric
progression starting from n = 0.
{ arn
}= a, arn, arn+1, arn+2, ……
{ arn
}= 3, 3(2)1, 3(2)2, 3(2)3, 3(2)4,
……
{ arn
}= 3, 6, 12, 24, 48, ……
Therefore,
the geometrical progression { arn } is 3, 6, 12, 24, 48, ……
Arithmetic Progression
Arithmetic
Progression is a sequence with a pattern like a, a + nd, a + (n + 1) d, a + (n
+ 2) d, ……, where a is the intial term of the sequence and d is the common
difference. Each term in arithmetic progression has the same difference with
the terms next to it.
Example 4:
Consider an = 1 + 3n. Please write down the
sequence of {an} starting from n = 1.
Solution:
a1
= 1 + 3 (1) = 4
a2 =
1 + 3 (2) = 7
a3
= 1 + 3 (3) = 10
a4
= 1 + 3 (4) = 13
a5
= 1 + 3 (5) = 16
Therefore
the sequence for arithmetic progression, {an} is 4, 7, 10, 13, 16,
……
Recurrence Relation
Recurrence
relation is a way of expressing each term in a sequence by referring to its
previous terms before the specified term.
Example 5:
Given an
= an-1 + 2. Find {an} starting from n = 1, if a0
= 1.
Solution:
From the
given statement an = an-1 + 2, In order to find an-1,
we have to determine the previous term for an, which is an-1. To
determine an-1, we need its previous term too, which is an-2.
This method of solving a term by using
its previous term depicts the recurrence
relation.
Since the
question stated starting from n=1
start with a1. As the statement an = an-1 + 2 had
mentioned, a1 should be a1 = a0 + 2. Since a0
= 1 is given, therefore a1 = 1 + 2 = 3.
For a2,
a3 and the rest of the terms, we just carry out the same process as
what we did to find a1.
a2
= a1 + 2
a2
= 3 + 2
a2
= 5
a3
= a2 + 2
a3
= 5 + 2
a3
= 7
a4
= a3 + 2
a4
= 7 + 2
a4
= 9
The
sequence, { an } = 3, 5, 7, 9, …
Initial condition
An initial condition of a sequence with recurrence relation states the first term that would occur as the starting
point of the sequence. Refer to
Example 5, as we can see that the question stated if a0 = 1. The
expression a0 = 1 is the
initial condition for the sequence { an }. This means that the
sequence { an } starts with the term a0, which is 1. Then
followed by a1, a2, a3, a4, …… We
can also say that a0 is the starting term of the sequence { an
}
Special number sequences
Special
number sequences are derived from arithmetic sequence or geometrical sequence,
or sometimes both. The terms in a special number sequence are arranged in such
a way it can indirectly represents a mathematical
theorem or a formula.
There are
many examples that show the characteristics of a typical special number
sequence. Let us look at some of them.
Squared Number Sequence
As the name
of the sequence has told, a squared number sequence consists of the list of squared terms. Each term is represented by the form of n2 for any real number
Example 6:
Write a
squared number sequence, { an } for n = 1, 2, 3, 4, 5, ……
Solution:
{ an
} = { n2 }
{ n2
} = 12, 22, 32, 42, 52,
……
{ n2
} = 1, 4, 9, 16, 25, ……
This
special number sequence had indirectly showed us the theorem of squared numbers
and showed the squared numbers starting from n = 1 in a sequence.
Another
example of special number sequence is Fibonacci
sequence.
Fibonacci Sequence
A Fibonacci
sequence is a sequence in which obeys the initial
condition that f0 = 0 and f1 = 1 (We use f to represent terms in a Fibonacci
sequence). The rest of the terms is just merely the addition of the previous
2 terms before the
appointed term.
Example 7:
Write a
Fibonacci sequence, fn
from n = 2 to n = 6.
Solution:
The initial
condition for a Fibonacci sequence is f0
= 0 and f1 = 1. As the
definition had defined, a term is the sum of 2 previous terms before it.
Therefore, to find f2, we
have to sum up f1 and f0.
f2 = f1 + f0
In the
initial condition, it is given that f0
= 0 and f1 = 1.
f2 = 1 + 0
f2 = 1
We use the
same technique to find other consequtive terms.
f3 = f2 + f1
f3 = 1
+ 1
f3 = 2
f4 = f3 + f2
f4 = 2
+ 1
f4 = 3
f5 = f4 + f3
f5 = 3
+ 2
f5 = 5
f6 = f5 + f4
f6 = 5
+ 3
f6 = 8
Therefore,
the Fibonacci sequence, { fn
} from n = 2 = 1, 2, 3, 5, 8, ……
Lucas sequence
A lucas
sequence works similarly like a Fibonacci sequence. The differences between 2
of them is the initial condition, in which in Fibonacci
sequence, f0 = 0 while f1 = 1 but L0 and L1 can be of any
number when we are dealing with
Lucas sequence. (L is used to
represent a Lucas sequence.).
Example 8:
Construct a
Lucas sequence, Ln with
the initial condition, L0 = 4 while L1 = 2.
Solution:
As
mentioned earlier, Lucas sequence pays no heed to the value of f0 and f1 as they can be of any value.
We’ll be
using the same technique as what we’ve done in constructing the Fibonacci
sequence.
To find f2, we find the sum of 2
previous terms before it. which is f1 and f0.
L2 = L1
+ L0
Given that f0 = 4 and f1 = 2.
L2 = 2 + 4
L2 = 6
Same
technique used for other consecutive terms.
L3 = L2 + L1
L3 = 6
+ 2
L3 = 8
L4 = L3 + L2
L4 = 8
+ 6
L4 = 14
L5 = L4 + L3
L5 = 14
+ 8
L5 = 22
Therefore,
the Lucas sequence, { Ln }
from n = 2 = 6, 8, 14, 22, ……
Use of Sequence in Computer Programming
Sequence is
very useful in the field of programming.
Example 9:
Every year, 10%
of students graduated from SPM is chosen arbitrarily to must attend the
National Service, which we usually called it as PLKN.
You are asked
to write a program for PLKN in which the program can choose arbitrarily the
participants for them.
Solution:
We can use the
concept of sequence here. For example, the list of all SPM students are key in
in the database. Each of them has a unique ID. Therefore, the first student in
the list might get an ID of 1 while the second student gets an ID of 2. Same as
the rest of them. The sequence of all the students ID listed should look like
this:
1, 2, 3, 4, 5, 6, ……
We take the
first 10 students in the list for example:
ID
|
Name
|
1
|
Alex Soh Chye Wat
|
2
|
Chew Xiang Leng
|
3
|
Goh Yoon Kiat
|
4
|
Fatimah Che Bakri
|
5
|
Muhammad Kushairi
|
6
|
Chee Bai Kia
|
7
|
Lim Lao Bu
|
8
|
Goh Song Chiang
|
9
|
Karshan a/l Rajendran
|
10
|
Nanthini a/p Gunasegaran
|
We can make a Java
program that reads the input from the user.
For example, a
user input 2 as the number. Therefore, all the students with the ID which is
divisible by 2 are chosen to enter PLKN. The ID which can be divisible by 2 can
be presented in a sequence form like this:
2, 4, 6, 8, 10,
……
Therefore,
refer to the table of ID and students’ name, we can know that Chew Xiang Leng,
Fatimah Che Bakri, Chee Bai Kia, Goh Song Chiang and Nanthini a/p Gunasegaran
are chosen for the PLKN programme.
If the user
inputs 3, means that the ID 3, 6, 9, …… would be chosen. Refer to the table,
Goh Yoon Kiat, Chee Bai Kia and Karshan a/l Rajendran are chosen for PLKN.
Mathematical Induction
Suppose
that we have a skyscraper with infinite height. We need to determine if we are
able to reach all of the floors of the skyscraper.
Let P(n) =
We can reach the nth floor of the skyscraper.
To prove
that ∀ n P(n) is true, we have to go through
all the floors to prove it. Since the skyscraper has infinite height, there are
infinite floors in the skyscraper. We can’t go through each floor to proof ∀ n P(n). In
this case, we have to use another more simpler way to proof it.
We know
that, to reach the first floor, we must go through the ground floor first. Same
as any other floors of the skyscraper. For example, if we want to reach the 37th
floor, we must access the 36th floor first. So as the 201st
floor, we have to get through the 200th in order to reach 201st.
From the
above examples, we can deduce a statement like this:
Q (n) =
We’ve to go through the nth floor first before we can go to the (n +
1)th floor.
Since Q(n)
is true for all the floors in the skyscraper, therefore we can proof that ∀ n P(n) is true, which is we can access to each of the floors after we’ve
accessed the previous floor.
The
technique that we’ve used in the above example to determine if all the values
are true or not, is called mathematical
induction.
The
Method Of Proof By Mathematical Induction
Let a statement,P be true with any
natural number, n.
To prove the statement by induction, there
are 2 steps that we need to do:
1. The basis step:
Initially
we assume n = 1. We prove P(1) is true.
2. The inductive step:
After the basis step, we prove that
if P(n) is true, then P(n + 1) is true. While n represents
any natural numbers.
After
the above 2 steps, we are able to prove that P(n) is true for all values of n.
Because there are technical errors when doing with the example part, so please view the example from this link:
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