SOME MATHEMATICAL OBSERVATIONS: Part 1 "How I squared it?"

I usually do not talk about my mathematical abilities. People close to me, often talk about my quick calculating abilities. To put it simply, I would say, it’s nothing special that I do. I calculate each calculation, and that’s about it. Yes, I like to research on the mathematical calculations, and so here I am, sharing with the readers, something unique. I am not sure, if the following two have been researched or not, and so I do not lay any claims on it. True to my words, I am not circulating my blog amongst my readers. I call my research work an observation more than anything else.

SQUARE of a number:
We all seem to be fascinated by squaring of a number, especially when we are young and at a later stage, preparing for some competitive exam or the other. This particular observation was done by me when I was probably in my first year of graduation. It may seem to be similar to the FIBONACCI SERIES, though it is unique in its own way.

The square of a number can be found out by adding the number of which the square is to be determined and the number preceding it, to the square of the number preceding it. So, let us assume the following:
A= 1st number.
B= 2nd number.

Then the square of “B” is given by:
Bsquared=(A+B) + Asquared

If, the same can be put in the form of a mathematical equation, the same can be re-written as:
N2(squared) = {N + (N-1)} + (N-1)squared Eq (i)

Let me consider two numbers: 2&3 as an illustration to this observation:
N=3
(N-1)=2
(N-1)2=4

We need to calculate N2(squared) i.e 3squared
Now, N+(N-1) = (2+3) = 5

So, substituting the above values in Eq (i), we get the following results:
3squared = (5+4) = 9

Similarly if we take another set of two numbers i.e 10 & 11.
N=11
(N-1)=10
(N-1)2=100

We need to calculate Nsquared i.e 11squared
Now, N+(N-1) = (11+10) = 21

So, substituting the above values in Eq (i), we get the following results:
11squared = (100+21) = 121

This series continues for the calculation of the squares of all the following numbers running into millions & billions and so on and so forth. As a result of this, the tedious task of doing the calculations or having to remember the squares of a number can be let go, by making use of this observation. In the following blog, I will describe yet another research observation, wherein I will describe a linkage in the calculation of the cube of a number.