Let’s look at a very simple number pattern, right..? Let’s say 1, 3, 5, 7… What are these if you look at them? What are they? These are nothing but odd numbers? These are nothing but odd numbers, right..?
Now you could visualize them as triangles..? One triangle… This is three triangles… If I divide it further, it’ll be five triangles here… Seven triangles… right?
So if I want to divide a triangle into each part, into equal areas, the number of triangles I will get in each row is going to be an odd number, right? This is one way to visualize it.
Now let’s add them and see… One… 1 and 3 you add, you’ll get 4… 1 + 3 + 5 you’ll get 9… 1 + 3 + 5 + 7 you’ll get 16… right? What are these..? Nothing but square numbers, right? Why do you get square numbers ? Because, adding triangles gives you squares, right ?
Two triangles make up a square..! Which is why adding these numbers that represent a triangle gave you a square..! So it’s just one way to see it… right?
There is another way we can visualize triangles… Let’s take 1… Then I have 1, … and 2, … 1, 2 and 3, … 1, 2, 3 and 4, … These are all triangles, right? The first one has just one point… The second one has three points… The third one has six points.. The fourth one has ten points… right? So we call all of these as triangular numbers, right? We call them as triangular numbers, because as you see they form triangles… right? Now, again we say if we add a triangle you get squares… So if you add these triangles you should again be getting squares… right? Let’s try… We have one first… 1 and 3 gives you 4 … 3 and 6 gives you 9 … Square again… 6 and 10 gives you 16… Square again… right? So you get these square numbers..! These are nothing but square numbers… Right? 1, 4, 9, 16 are nothing but square numbers… right? You can see that in the figure also… What we did here..? 1 + 3 … So, 1 and 3 gave you 4 … right? I’m just rotating the triangle and putting it on top… 3 and 6 gives you 9… right? 6 and 10 gives you 16… right?
So multiple ways in which you can visualize a simple concept of triangles and squares… right? It actually links numbers, geometry, algebra, … everything… right? Which is why number patterns are so fascinating… right? When you start visualizing number pattern, they become extremely fascinating !