Mathematical Induction: Simple Problem

Brief Explanation: Today we explore a part of mathematics that I don’t fully grasp, in hope that my comprehension gets better after trying to explain it to you. This is just a simple exercise to get us started.

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Question: Prove the following formula:

enunciado.png— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —

Comprehending: We are going to use mathematical induction to prove this formula. I’m not going to explain here the principle of mathematical induction in detail, but the gist is that:

Suppose we have some property P that holds for a certain number x. We can generalize this property to all natural numbers x if the two following conditions are met:

(1) P(1) is true

(2) Whenever P(k) is true, P(k+1) is true.

(This definition is presented in Michael Spivak’s Calculus, I inserted it here because I believe it explains very briefly what mathematical induction is.)

So the steps we have to do are: prove the property for P(1) and then for P(k+1). If you think about this, the mental image of the domino pieces makes sense: with k+1, you’ve proved the property for all numbers.

Another thing we’ll need is a very useful formula. The sum of the n first natural numbers is:

somatorio n 1º termos.png

This is a formula you should keep in mind, because it comes in handy in many situations, and we’ll use it here.

Now, let’s get to the actual exercise.

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Solving: First, let’s prove the formula for n=1. This is very easy:

prova 1.png

Generally, proving the property for P(1) is very easy. In this case it was just a simple substitution and the result was immediate.

Now, what do we want to prove? Basically, we need to prove the formula given to us holds for n+1. Expressing this mathematically:

a provar.png

Lets works on what we want to prove. If we define:

3.png

We can simplify our condition:

4.png

Developing this expression:

5.png

You probably have already noticed this: what exactly is AA is the sum of the n first natural numbers! And what is A^2? A^2 is the same as (1+…+n)^2, and by the initial formula, is the same as (1^3+…+n^3)!

Applying the formula we discussed in the Comprehending section and this notion about A^2:

6.png

Now all we got to do is to use the common terms in the expression:

7.png

And it’s done! We proved the formula!

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Conclusion: I’ll probably keep posting these problems about proofs and induction. Although I feel it is an area I’m clearly not comfortable with, writing these posts makes me think about these subjects and deepens my comprehension (at least I hope)!

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