A Beautiful Geometric Argument

Most proofs in mathematics tend to be very time consuming, involving a lot of complex steps and tedious, making many people wary of them. Sometimes, there are alternate methods for proving the same which are pretty straightforward and simple. You might tend to be blown off by its simplicity and remark “Is that all?”

Today, I will be sharing one such proof, which, if done using traditional algebra tends to become very messy and calculation intensive, but, can be greatly simplified when approached using geometry.

The proof I am going to present today is the proof for the formula of the sum of cubes of the first n natural numbers. You can go through the algebraic proof here. I suggest you go through it first so that you can appreciate the simplicity of the proof I am going to present below

(I read this geometric proof in Quora a long time back and thought of sharing it with all of you. Since its been a long time, I am unable to trace the link for the Quora answer. However, I will reproduce the same here to the best of my abilities)

What We Need To Prove?

We need to show that:

The only formulae we will be needing in this proof are 2 basic formula, which I guess all of you are well aware of:

Required Things

The things that we will be requiring for this proof are the 2 formulae I have mentioned above and a square which is built out by joining a series of gnomons(see below)

The resultant square which we need is:

Let the total area of the square be A and the area of the gnomons be G1, G2, …., Gn

The Proof

The length of the side of the square is 1+2+…+n

Hence,

Another way to calculate the area of the square is by summing the areas of the constituent gnomons.

G1 is a square of side 1. Hence, its area is 1

The area of G2 is the area of the bigger square(length of side 1+2=3) minus the area of the square of side 1 i.e G1

The area of any gnomon is equal to the area of the bigger square(of side length 1+2+…n) minus area of the smaller square(of side length 1+2+….n-1). The area of the kth gnomon can be written as:

Equating the equations 1 & 2, we get

Hence proved!!!

We have got the required result. Did we do any complex calculations? No. Did we do any complicated algebra? No. Using a few basic formulae and a square, we arrived at the formula in a very simple way. Those who went through the algebraic proof will appreciate the simplicity of this geometric proof.

There exist many such proofs in mathematics which have a shortcut which greatly simplifies things. Will try to present more such simple proofs if I come across them in the future

Thanks for reading!!!

Originally published at http://infinitesimallysmallcom.wordpress.com on May 3, 2020.