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Factoring (n + 1) out, we get

      (n + 1)(n + 2) / 2,

which is equal to the RHS for n+1.

Thus LHS = RHS for n+1.

End of Proof.

Example of use of mathematical induction --- program correctness

Loops in an algorithm/program can be proven correct using mathematical induction. In general it involves something called "loop invariant" and it is very difficult to prove the correctness of a loop. Here we are going to give a few examples to convey the basic idea of correctness proof of loop algorithms.

First consider the following piece of code that computes the square of a natural number:

(We do not compute the square this way but this is just to illustrate the concept of loop invariant and its proof by induction.)

SQUARE Function: SQ(n)

S<- 0

i<- 0

while i<n

    S<- S + n

    i<- i + 1

return S

Let us first see how this code computes the square of a natural number. For example let us compute 3 2 using it.

First S<- 0 and i<- 0 give S = 0 and i = 0 initially.

Since i<3, the while loop is entered.

    S<- 0 + 3

    i<- 0 + 1

producing S = 3 and i = 1.

Since i<3, the while loop is entered the second time.

    S<- 3 + 3

    i<- 1 + 1

producing S = 6 and i = 2.

Since i<3, the while loop is entered the third time.

    S<- 6 + 3

    i<- 2 + 1

producing S = 9 and i = 3.

Since i = 3, the while loop is not entered any longer, S = 9 is returned and the algorithm is terminated.

In general to compute n2 by this algorithm, n is added n times.

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 0 and n is a natural number. Thus i eventually becomes equal to n.

Next, to prove that it computes n2, we show that after going through the loop k times, S = k*n and i = k hold. This statement is called a loop invariant and mathematical induction can be used to prove it.

Proof by induction.

Basis Step: k = 0. When k = 0, that is when the loop is not entered, S = 0 and i = 0. Hence S = k*n and i = k hold.

Induction Hypothesis: For an arbitrary value m of k, S = m * n and i = m hold after going through the loop m times.

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

    S<- m*n + n

    i<- i + 1

producing S = (m + 1)*n and i = m + 1.

Thus S = k*n and i = k hold for any natural number k.

Now, when the algorithm stops, i = n. Hence the loop will have been entered n times.

Thus S = n*n = n2. Hence the algorithm is correct.

The next example is an algorithm to compute the factorial of a positive integer.

FACTORIAL Function: FAC(n)

i<- 1

F<- 1

while i<= n

    F<- F * i

    i<- i + 1

return F

Let us first see how this code computes the factorial of a positive integer. For example let us compute 3!.

First i<- 1 and F<- 1 give i = 1 and F = 1 initially.

Since i<3, the while loop is entered.

    F<- 1 * 1

    i<- 1 + 1

producing F = 1 and i = 2.

Since i<3, the while loop is entered the second time.

    F<- 1 * 2

    i<- 2 + 1

producing F = 2 and i = 3.

Since i = 3, the while loop is entered the third time.

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
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
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
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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Discrete structures. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10768/1.1
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