Visualising a Maths Problem
Day 6 / 365
Let me present you with a math problem which a lot of you might have seen in school. If you hate maths don’t leave this page just yet, let me explain !
Find the sum of first n odd numbers
The problem statement is quite simple. Given a number n, find the sum of first n odd numbers. So
For n=1 the answer is 1
For n=2 the answer is 1 + 3 =4
For n=3 the answer is 1 + 3 + 5 = 9
For n=4 the answer is 1 + 3 + 5 + 7 = 16
Do you see a pattern yet? I’ll give you a hint, here are the squares of some integers
1 squared is 1 X 1 = 1
2 squared is 2 X 2 = 4
3 squared is 3 X 3 = 9
4 squared is 4 X 4 = 16
That’s some progress! so it seems that the sum of first n odd numbers has to be n squared. But can we proof it without using any complex equations?
Let’s give it a go, take a look at the figure below. We have 4 squares of different colours overlapping each other.
The smallest white 1 X 1 square at the bottom left has 1 tile in it.
The next 2 X 2 square has 3 more tiles than the 1st one, that’s 4.
And the next 3 X 3 square has 5 more tiles than the previous one, that’s 9
Each next square is bigger than the previous one by an odd number of tiles!
And we know that the number of tiles in a n X n square is n squared (which is why we call it n squared)
That’s it! This proof might not get you full marks in an exam, but I hope this can convince you that maths is more than just abstract equations.
This story is part of my 365 Day Project for 2019. Read about it here