Mathematical Proof By Induction

Proof by induction is just one type of mathematical proof that follows a main method:

1) BASIS:
Prove the general statement is true for n = 1

2) ASSUMPTION:
Assume the general statement is true for n = k

3) INDUCTIVE:
Show that the general statement is then true for n = k + 1

4) CONCLUSION:
The general statement is then true for all positive integers, n.


And an example question:

(BASE STEP)
Prove by the method of mathematical induction that for n is a set of positive natural numbers:

Sum from r=1 to n; (2r-1) = n^2

n=1; LHS = 2(1)-1 = 1
        RHS = 1^2 = 1
Therefore true for n = 1.

(ASSUMPTION)
Assume that the summation formula is true for n = k;

Sum from r=1 to k (2r-1) = k^2

With n = k + 1, terms the summation formula becomes:

(INDUCTIVE STEP)
Sum from r = 1 to (k+1) of (2r-1) = 1+3+...+(2k-1)+(2k+1)
                                                   
= k^2                 + (2k+1)
=k^2 + 2k + 1
=(k+1)^2

Therefore summation formula is true when n = k + 1

(CONCLUSION)
If the summation formula is true for n = k then it is shown to be true for n = k + 1. As the result is true for n = 1, it is now also true for all n  ≥ 1 and n is a set of positive natural numbers by mathematical induction.

TIPS
When trying to prove the inductive step, a very good idea is to write the formulae out that you have derived^^ i.e. (k^2 + 2k + 1) and then write out the other part of the formula (k^2) but replacing k with the summation i.e. k+1; therefore you get (k+1)^2.
Then it is much easier to prove the formulae as you have (k^2 + 2k + 1) and now it is just a matter of rearranging formulae to get (k+1)^2; which in this case is very easy, but helps a lot when the questions become harder.

Cambridge CUSU Shadowing Scheme

A few weeks ago I attended the Cambridge CUSU Shadowing Scheme, which allowed me to stay at Kings College Cambridge for 3 days, where I would sit, eat, study and socialise with the current undergraduates at the College.

Obviously I shadowed Mathematics while I was there; attending 3rd and 4th year lectures such as Set Theory and Geometric Group Theory (very complex maths).

One of the first things that I was so astonished by when I arrived was the amount of people that rode bicycles! There were hardly any cars on the road, as they were filled with cyclists. Now I realise that Cambridge, being a relatively small campus was mostly accessible by bicycle and therefore it would be much more efficient for people to cycle; for increased speed and saving money.

Meeting 1st, 2nd, 3rd and even 4th year students from Cambridge; it has really opened my eyes to the actual size of mathematics and how inexperienced I am. I feel that there is infinite amounts to be learnt and still many more ideas to be revealed.

I liked the idea that it was also possible for students in the first year, and so on to attend lectures that they are not currently studying. For example: my mentor is currently in his first year at Cambridge, but attended lectures that were in the third and even fourth year (that is degree mathematics!). I felt that this; along with the absolutely enormous library that the Colleges had; really allowed students to maximise their productiveness in study time and allow them to develop an understanding of the topic that far outweighed the average.

From what I had gathered, I also found that the connection between students that were not at the same college was great. As they all have the motivational drive to work hard in their given subjects, it was easy for people to spark conversation and discuss complex theories that they had been recently studying.

Concluding; Cambridge is awesome!

A Level Maths Lecture - Dr Piers Bursill-Hall

Earlier today, I attended a very interesting A Level Maths Lecture with the speakers Dr Piers Bursill-Hall The talks were very interesting and gave a wide range of insights into the uses and application of mathematics; from how the Heliocentric Model was proposed and developed.

Dr Piers Bursill-Hall

Being Cambridge University's most entertaining Maths Lecturer; his talk was very engaging. He commented on how "Copernicus was wrong, and that even his mathematics was wrong." 

Firstly, he announced that Copernicus was most definitely not the first person to invent the Heliocentric Model, and that really he was one of many to have looked at it by his time. It was said that Copernicus also did not only have one theory on the model; but actually he had two. He stated that the first model was that the physical representation of the solar system was with the Sun in the centre, and that all the planets orbited around the Sun, with the stars furthest away. Then there was the second model that went into mathematical detail into how the planets and moons were not actually in a singular elliptical orbit at all, but orbits upon orbits upon orbits. It was said that in the model there was roughly 150 different orbits! 

Finally, the Romans had started to believe in the idea of Christianity. However it was easy to make a link between both the Sun being in the centre of the Solar System and God. An analogy was described that if you imagine a man living in a cave. He has lived there for his whole life and has not been exposed to much light at all. Suddenly the man is exposed to the outdoors and he becomes blinded by the suns light. As the man becomes to adapt to the new surroundings and be able to see things, we will see shapes, flowers, grass and so on. Eventually he will be able to see the sun, and that the sun is the last thing that he will see because it brings light to everything else and it is the most powerful being. This related the centre of the Heliocentric Model to be the most powerful, and now the link between God and the Sun had been made. 

A Level Maths Lecture - Marcus du Sautoy

Marcus du Sautoy,

An Oxford Professor for the Public Understanding of Science, Professor of Mathematics and co-host of the TV show "School of Hard Sums".

Having already purchased some of his books, "Does God play dice" and "Finding Moonshine", I have found an interest in Chaos or Clear Cut and how the events that we encounter are a result of different actions.
He spoke showed us a contraption, that I think that he made. It was a small; two pieced rod that was attached to a plank of wood, and had two pivots, acting almost as a leg, with hip and knee pivots. The contraption showed that given different amounts of energy and height that he pushed the device with, it would react extremely differently. This was the introduction to the idea of 'Chaos'.
He exposed us to the life cycle of Lemmings. They are small creatures and have an unusual life span where every few years, the population massively decreases. He used the idea of Chaos to show that the lifespan of the lemmings was completely random and it was a mystery of why the lemmings suddenly would all die out. It turns out the BBC made a documentary - The 1958 Walt Disney documentary, White Wilderness, Part II, that showed the BBC recording the lemmings jumping off cliffs and committing suicide; and this was the reason to their fall in population. However it turns out that it was all fake and that this was not the natural nature of the lemmings at all. He also spoke about another type of creature that died out every 17 years. Mathematicians had wondered why many creatures all had relationships with life cycles and prime numbers, such as 17. Eventually it was theorised that their life cycle was strategically planned to avoid contact with predators. It was shown that if the predator returns every 3 years to where the creatures live, it would take 51 years for the two species to cross paths, and therefore gives them a higher chance of survival, than say if they had a life cycle of 6 years, where they would meet every 6 years. This idea of creatures using prime numbers as a survival instinct was very interesting.
Finally he spoke about Football. What is a lecture without football! He mentioned the World famous free kick goal by Roberto Carlos vs France in 1997. He spoke about the trajectory of the curve of the ball and how there is mathematics behind it. He said that the ball, as you may have sometimes seen, had the appearance of being weightless, and floating while travelling. This was because the force was so great on the ball, that Chaos was actually occurring and the ball was not able to spin, and therefore travelled in a straight line. However, when the trajectory of the ball had reached a certain point; the forces and spin on the ball kicked in, the the fall reared off to the left at the last minute, scoring in the bottom right corner; leaving the goalkeeper and everyone else who saw it shocked.