These slides contain proof of Binomial theorem , Tower of Honai , recursion a d properties of relation...

1. STARTS WITH THE NAME OF
ALLAH
the most beneficent & Merciful

2. PRESENTAION
AQEEL RAFIQUE 01
HAMZA MAQSOOD 12
REYAN IQBAL 28
UMAIR HAIDRI 17
SHOAIB ASHRAF 44
RAJA AMIR 43
PRESENTED BY :

3. Binomial Theorem

4. Binomial :
In algebra (a+b) is called binomial.
Binomial theorem provides an expression for the power of
binomial of n.

5. Binomial Theorem
The expansion of binomial theorem is.
푎 + 푏 푛 =
푛
푘=0
푛
푘
푎푛−푘 푏푘
푎 + 푏 푛 =an+
푛
1 푎푛−1푏1 + 푛
2 푎푛−2푏2 + ⋯ + 푛
푛−1 푎1푏푛−1+bn

6. Binomial Theorem
Need of Binomial Theorem
As we know
(a+b)0 =1
(a+b)1=(a+b)
(a+b)2 =a2+2ab+b2
.
.
(a+b)n=?

7. Binomial Theorem
푎 + 푏 0 =
0
푎 + 푏 푛 =
푘=0
푛
푘=0
Proof by Algebraic method
0
푘 푎0−0푏0 = 1
Let (1) is true for n=m
푎 + 푏 푚 =
푛
푘 푎푛−푘푏푘 (1)
푚
푘=0
푚
푘
푎푚−푘푏푘
If true for n=m then also true for n=m+1

8. Binomial Theorem
푎 + 푏 푚+1 =
푚+1
푘=0
푚+1
푘 푎푚+1−푘푏푘
R.H.S
푎 + 푏 푚+1 = 푎 + 푏 푎 + 푏 푚
=(a + b)
푚
푘=0
푚
푘 푎푚−푘푏푘
=a .
푚
푘=0
푚
푘 푎푚−푘푏푘+b .
푚
푘=0
푚
푘 푎푚−푘푏푘
=
푚
푘=0
푚
푘 푎푚+1−푘푏푘+
푚
푘=0
푚
푘 푎푚−푘푏푘+1
Replacing variables:
Let j=k+1; k=j-1 when k=0, j=1
when k=m, j=m+1

9. Binomial Theorem
푎 + 푏 푚+1 =
푚
푘=0
푚
푘 푎푚+1−푘푏푘 +
푚
푘=0
푚
푘 푎푚−푘푏푘+1
Taking second summation on the right hand side.
=
푚
푗−1 푎푚−(푗−1)푏푗−1+1
=
푚
푗−1 푎푚+1−푗)푏푗
=
푚+1
푗=1
푚
푗−1 푎푚−푗+1푏푗
=
푚+1
푘=1
푚
푘−1 푎푚+1−푘푏푘
푎 + 푏 푚+1 =
푚
푘=0
푚
푘 푎푚+1−푘푏푘 +
푚+1
푘=1
푚
푘−1 푎푚+1−푘푏푘
=am+1+
푚+1
푘=1
푚
푘 + 푚
푘−1 푎푚+1−푘푏푘+bm+1

10. Binomial Theorem
푎 + 푏 푚+1 = am+1+
푚+1
푘=1
푚+1
푘 푎푚+1−푘푏푘 +bm+1 by Pascal’s formula
푎 + 푏 푚+1 =
푚+1
푘=0
푚+1
푘 푎푚+1−푘푏푘
Hence proved.
This shows that if any number in the power of binomial is given we can easily
find its expansion.

11. Counting elements
in one dimensional
array.

12. Counting elements in one dimensional
array.
Let A[1],A[2],A[3]……………….A[n] is a one dimensional array. Where
n a positive integer.
To find the number of element in one dimensional array by using the
theorem.

13. Counting elements in one dimensional
array.
We use theorem of find the no of elements in a list.
i-e:- n-m+1
where n is the last term of the list and m is the first term of
the list.

14. Counting elements in one dimensional
array.
Example:
suppose the elements in 1 dimensional array are;
A[2]=2;
A[3]=3;
A[4]=5;
.
.
.
A[10]=7

15. Counting elements in one dimensional
array.
By Applying theorem we get
Apply theorem on index;
Where n=10, m=2;
The number of elements in the given array are:
n-m+1=10-2+1
=9
Elements = 9

16. Recursion

17. Recursion
Recursively Defined Sequence
Method of defining a sequence:
 Informal ways
 Explicit formula
 Recursion

18. Recursion
Recursively Defined Sequence
 Informal way:
In informal ways a sequence is given we extract or
generate the pattern of the sequence and generate the next
term.
 Disadvantages:
• Misunderstand of the sequence cause the error.
For example:
if the sequence 3,5,7……. Is given if some one
guess it prime number place 9 if someone understand it
prime number he put 11. so this cause the
misunderstanding.

19. Recursion
Recursively Defined Sequence
 Explicit formula:
In explicit formula we make a formula for the nth
term of the sequence.
For example: 2,4,6……….
Explicit formula for above sequence is 2k, where k>0
 Advantages:
• Each term is uniquely determine.
 Disadvantage:
• Difficult to make the explicit formula if such
sequence is given which is difficult to analyze.

20. Recursion
Recursively Defined Sequence
Recursion:
In recursion two equations are given.
• Recurrence relation:
It is the formula to find the sequence.
• Initials conditions:
it is the first few values of the sequence. It is also called
base or bottom of the recursion.

21. Recursion
Recursively Defined Sequence
 For example:
1) bk = bk-1+ bk-2 recurrence relation
2)b0=1, b1=3 initial conditions
Every founded value is used to find the next term of the sequence

22. For example:
A sequence c0, c1, c2, . . . recursively as follows: For all integers
k ≥ 2,
(1) ck = ck−1 + k.ck−2 + 1 recurrence relation
(2) c0 = 1 and c1 = 2 initial conditions.
Find c2,c3.
Recursion
Computing Terms of a Recursively Defined Sequence

23. Computing Terms of a Recursively Defined Sequence
Solution:
Recursion
Putting k=2
since c1 = 2 and c0 = 1
c2 = c1 + 2c0 + 1
= 2 + 2·1 + 1
=5
similarly for c3 we put k=3 and solve.

24. Recursion
Sequences That Satisfy the Same Recurrence Relation
Let a1, a2, a3, . . . and b1, b2, b3, . . . satisfy the recurrence relation
that the kth term equals 3 times the (k − 1) term for all integers
k ≥ 2
(1) ak = 3ak−1 and bk = 3bk−1
And the initial conditions are:
a1=3, b1=1
Find a2, a3 and b2 ,b3

25. Recursion
Sequences That Satisfy the Same Recurrence Relation
Solution:
a2 = 3a1 = 3·3 = 9
a3 = 3a2 = 3·9 = 27
So the ‘a’ sequence is 3,9,27…….
b2 = 3b1 = 3·1 = 3
b3 = 3b2 = 3·3 = 9
So the ‘b’ sequence is 1,3,9,………

26. Tower of Hanoi
The Tower of Hanoi (also called the Tower of
Brahma or Lucas' Tower, and sometimes
pluralized) is a mathematical game or puzzle. It
consists of three rods, and a number of disks of
different sizes which can slide onto any rod. The
puzzle starts with the disks in a neat stack in
ascending order of size on one rod, the smallest at
the top, thus making a conical shape.

27. Tower of Hanoi
A B C
1
n-1
n

28. Tower of Hanoi
Objective
The objective of the puzzle is to move the entire stack to
another rod, obeying the following simple rules:
• Only one disk can be moved at a time.
• Each move consists of taking the upper disk from one of
the stacks and placing it on top of another stack i.e. a disk
can only be moved if it is the uppermost disk on a stack.
• No disk may be placed on top of a smaller disk.

29. Recursive pattern:
From the moves necessary to transfer one, two, and three disks, we can
find a recursive pattern - a pattern that uses information from one step to
find the next step - for moving n disks from post A to post C:
First, transfer n-1 disks from post A to post B. The number of moves will
be the same as those needed to transfer n-1 disks from post A to post
C. Call this number M moves.
Next, transfer disk 1 to post C [1 move].
Finally, transfer the remaining n-1 disks from post B to post C. [Again, the
number of moves will be the same as those needed to transfer n-1
disks from post A to post C, or M moves.]
No of moves : (n-1)+1+(n-1)
2(n-1)+1

30. Tower of Hanoi
What?
if we want to know how many moves it will take to transfer 100 disks
from post A to post B.
Ans: Through recursion we will first have to find the moves it takes to
transfer 99 disks, 98 disks, and so on.
So now we find the explicitly:
Number of Disks Number of Moves
1 1
2 3
3 7
4 15
5 31
The pattern generated from this sequence is: 2n-1

31. Tower of Hanoi
1,3,7,15…………….
1+0=1 21-1=1
1+2=3 22-1=3
3+4=7 23-1=7
7+8=15 24-1=15
. .
. .
. .
So the minimum number of moves required to solve a
Tower of Hanoi puzzle is 2n - 1, where n is the number of
disks.

32. Relations

33. Relations
A relation R is a subset of the Cartesian product of the
given set(s).
Given in order pair form (x , y).
x related to y, if and only if (x , y) is in R.
Denoted by x R y.

34. Relations
Properties of Relations:
 Reflexive
 Symmetric
 Transitive

35. Relations
Properties of Relations:
Let R be a relation on set A
 Reflexive:
R is reflexive if and only if x R x for all x is in A.
 Symmetric:
R is symmetric if x R y then y R x ; x , y is in A
 Transitive:
R is transitive if x R y and y R x then x R z; x , y , z is in A

36. Relations
Properties of Relations:
Let A = {0, 1, 2, 3} and define relations R on A as
follows:
R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)},
Is R reflexive, symmetric, transitive ?
Solution:
Graph of relation will be

37. Relations
Properties of Relations:
Reflexive:
R is reflexive because each element contain loop, mean each element is
related to itself
Symmetric:
R is symmetric because here an arrow move from one point to second and
also from second to first, mean first related to second and also second
related to first.
Transitive:
R is not transitive because there is no arrow moves from 3 to 1. so 1 R 0 and
0 R 3 but 1 R 3.

38. Any
Question?

39. Thanks
!