Why the differences between each number of a $x^2$ sequence is always an odd number and increase by $2$?

Question

I'm not a math person at all and I realize that this might be obvious, I'm trying to increase my awareness about it, so please excuse me if the question is too basic. Also excuse my lack of formatting in expressing my ideas, any tip or correction would be appreciated.

If you square the elements of a sequence of natural numbers $(1, 2, 3, 4,...)$ you respectively get $1,4,9,16,...$ If you calculate the difference between each consecutive element, you get $3,5,7, ...:$

This list of differences would always be composed of odd numbers. Why?

Also, why does it 'grows' linearly, increased by $2$ on every step?

Thanks.

Answer 1

If $n\in\Bbb N$, then $(n+1)^2-n^2=2n+1$, which is an odd number. Actually, every positive odd number (other than $1$) can be obtained by this process. Besides, the sequence $(2n+1)_{n\in\Bbb N}$ grows linearly.

Answer 2

Here is a well known proof without words:

https://steemit.com/math/@ertwro/math-proofs-without-words-visually

Answer 3

When you square an even number, you are effectively adding up an even number of even number of times, which always results in an even number (eg. $4^2 = 4+4+4+4$). When you square an odd number, you are adding up an odd number an odd number of times, which always results in an odd number (e.g. $3^2 = 3+3+3$). The squares of even and odd numbers are also even and odd, and since even and odd numbers alternate, their squares and will also alternate being even or odd. The difference in consecutive squares must therefore be odd, since we can only change the parity of the square by adding an odd number (if $X^2$ is odd then $(X+1)^2$ is even, so the difference must be odd, and the same result is true if $X^2$ is even and $(X+1)^2$ is odd).

As for why the difference grows linearly, consider drawing a grid to represent a squared number like 3x3. To get to the next squared number, you need to add another row and column of squares - as you extend up and to the right, you'll need exactly 2 more squares than you needed last time. Below, the orange squares represent "what you added last time", while the green squares represent the extra 2 squares responsible for the growth in the difference of squares. To get to the next squared number, you always add "what you added last time" plus 2, resulting in the linear growth in the difference of consecutive squares.