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    F<- 2 * 3

    i<- 3 + 1

producing F = 6 and i = 4.

Since i = 4, the while loop is not entered any longer, F = 6 is returned and the algorithm is terminated.

To prove that the algorithm is correct, let us first note that the algorithm stops after a finite number of steps. For i increases one by one from 1 and n is a positive integer. Thus i eventually becomes equal to n.

Next, to prove that it computes n!, we show that after going through the loop k times, F = k ! and i = k + 1 hold. This is a loop invariant and again we are going to use mathematical induction to prove it.

Proof by induction.

Basis Step: k = 1. When k = 1, that is when the loop is entered the first time, F = 1 * 1 = 1 and i = 1 + 1 = 2. Since 1! = 1, F = k! and i = k + 1 hold.

Induction Hypothesis: For an arbitrary value m of k, F = m! and i = m + 1 hold after going through the loop m times.

Inductive Step: When the loop is entered (m + 1)-st time, F = m! and i = (m+1) at the beginning of the loop. Inside the loop,

    F<- m!* (m + 1)

    i<- (m + 1) + 1

producing F = (m + 1)! and i = (m + 1) + 1.

Thus F = k! and i = k + 1 hold for any positive integer k.

Now, when the algorithm stops, i = n + 1. Hence the loop will have been entered n times. Thus F = n! is returned. Hence the algorithm is correct.

Mathematical induction -- second principle

There is another form of induction over the natural numbers based on the second principle of induction to prove assertions of the form ∀x P(x). This form of induction does not require the basis step, and in the inductive step P(n) is proved assuming P(k)   holds for all k<n . Certain problems can be proven more easily by using the second principle than the first principle because P(k) for all k<n can be used rather than just P(n - 1) to prove P(n).

Formally the second principle of induction states that

      if ∀n [ ∀k [ k<n size 12{ rightarrow } {} P(k) ] size 12{ rightarrow } {} P(n) ] , then ∀n P(n) can be concluded.

Here ∀k [ k<n size 12{ rightarrow } {} P(k) ] is the induction hypothesis.

The reason that this principle holds is going to be explained later after a few examples of proof. Example 1: Let us prove the following equality using the second principle:

For any natural number n , 1 + 3 + ... + ( 2n + 1 ) = ( n + 1 )2.

Proof: Assume that 1 + 3 + ... + ( 2k + 1 ) = ( k + 1 )2   holds for all k,   k<n.

Then 1 + 3 + ... + ( 2n + 1 ) = ( 1 + 3 + ... + ( 2n - 1 ) ) + ( 2n + 1 )

= n2 + ( 2n + 1 ) = ( n + 1 )2 by the induction hypothesis.

Hence by the second principle of induction 1 + 3 + ... + ( 2n + 1 ) = ( n + 1 )2   holds for all natural numbers.

Example 2: Prove that for all positive integer n, i = 1 n size 12{ Sum rSub { size 8{i=1} } rSup { size 8{n} } {} } {} i ( i! ) = ( n + 1 )! - 1

Proof: Assume that

1 * 1! + 2 * 2! + ... + k * k! = ( k + 1 )! - 1   for all k,   k<n.

Then 1 * 1! + 2 * 2! + ... + ( n - 1 ) * ( n - 1 )! + n * n!

= n! - 1 + n * n!    by the induction hypothesis.

= ( n + 1 )n! - 1

Hence by the second principle of induction i = 1 n size 12{ Sum rSub { size 8{i=1} } rSup { size 8{n} } {} } {} i ( i! ) = ( n + 1 )! - 1   holds for all positive integers.

Example 3: Prove that any positive integer n, n>1, can be written as the product of prime numbers.

Questions & Answers

how can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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Source:  OpenStax, Discrete structures. OpenStax CNX. Jan 23, 2008 Download for free at http://cnx.org/content/col10513/1.1
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