Binomial Theorem, Recursion ,Tower of Honai, relations
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Transcript of Binomial Theorem, Recursion ,Tower of Honai, relations
STARTS WITH THE NAME OF ALLAH
the most beneficent & Merciful
PRESENTAION
AQEEL RAFIQUE 01
HAMZA MAQSOOD 12
REYAN IQBAL 28
UMAIR HAIDRI 17
SHOAIB ASHRAF 44
RAJA AMIR 43
PRESENTED BY :
GROUP MEMBERS :
Binomial Theorem
Binomial :In algebra (a+b) is called binomial. Binomial theorem provides an expression for the power of binomial of n.
Binomial TheoremThe expansion of binomial theorem is.
an++bn
Need of Binomial TheoremAs we know (a+b)0 =1 (a+b)1=(a+b)(a+b)2 =a2+2ab+b2
.
.(a+b)n=?
Binomial Theorem
Binomial Theorem
= 1Let (1) is true for n=m
If true for n=m then also true for n=m+1
(1)Proof by Algebraic method
Binomial TheoremR.H.S
=(a + b) =a .+b . =+
Replacing variables:Let j=k+1; k=j-1 when k=0, j=1
when k=m, j=m+1
Binomial TheoremTaking second summation on the right hand side.
====
=am+1++bm+1
Binomial Theorem by Pascal’s formula
Hence proved.This shows that if any number in the power of binomial is given we can easily find its expansion.
Counting elements in one
dimensional array.
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.
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.
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
Counting elements in one dimensional array.
By Applying theorem we getApply 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
Recursion
RecursionRecursively Defined Sequence
Method of defining a sequence: Informal ways Explicit formula Recursion
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.
RecursionRecursively 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.
RecursionRecursively 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.
RecursionRecursively Defined Sequence
For example:1) bk = bk-1+ bk-2 recurrence
relation
2)b0=1, b1=3 initial conditions
RecursionRecursively Defined Sequence
Every founded value is used to find the next term of the sequence
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.
RecursionComputing Terms of a Recursively Defined Sequence
Solution:Putting k=2 since c1 = 2 and c0 = 1
c2 = c1 + 2c0 + 1 = 2 + 2·1 + 1 =5similarly for c3 we put k=3 and solve.
RecursionComputing Terms of a Recursively Defined Sequence
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
RecursionSequences That Satisfy the Same Recurrence Relation
RecursionSequences That Satisfy the Same Recurrence RelationSolution:
a2 = 3a1 = 3·3 = 9a3 = 3a2 = 3·9 = 27
So the ‘a’ sequence is 3,9,27…….
b2 = 3b1 = 3·1 = 3b3 = 3b2 = 3·3 = 9
So the ‘b’ sequence is 1,3,9,………
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.
nn-1
1
A B C
Tower of Hanoi
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.
ObjectiveTower of Hanoi
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
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
Tower of Hanoi
Tower of Hanoi
1,3,7,15…………….1+0=1 21-1=11+2=3 22-1=33+4=7 23-1=77+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.
Relations
RelationsA 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.
RelationsProperties of Relations:
Reflexive Symmetric Transitive
RelationsProperties 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
RelationsProperties 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
RelationsProperties of Relations:
Reflexive:R is reflexive because each element contain loop, mean each element is related to itselfSymmetric: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.
Any Question?
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