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$1^2 = 1$, $2^2 = 4$, $3^2 = 9$, $4^2 = 16$, $5^2 = 25$, etc...

Looking at the difference between those square values, we get: 3, 5, 7, 9, etc...

The difference from one (integer) square to the next increases by 2 without fault (let's assume).

Why is that? Why is there that pattern of increases by 2? What is it due to? What is the source of it?

I can "see" squaring visually as the construction of an actual square and I have drawn subdivided squares within squares to see the pattern unfold, but I just don't understand how to explain that increase by 2; I can't trace it, essentially.

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    $\begingroup$ Hint. Can you calculate $(x+1)^2) - x^2$? $\endgroup$ – Ethan Bolker May 7 '15 at 16:36
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    $\begingroup$ $(n+1)^2-n^2=2n+1$ $\endgroup$ – hamid kamali May 7 '15 at 16:36
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    $\begingroup$ It's very much of the same origin as the second derivative of $f(x)=x^2$. $\endgroup$ – Jyrki Lahtonen May 7 '15 at 16:55
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    $\begingroup$ @JyrkiLahtonen, sorry I don't see it, do you mind explaining the connection? $\endgroup$ – jeremy radcliff May 7 '15 at 16:59
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    $\begingroup$ Not him, but a derivative is infinitesimal difference whereas what you are looking at finite differences. $\endgroup$ – David Etler May 7 '15 at 22:12
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For a visual intuition.... The inner blue region increments the gray square by one, but to make the next increment you need the extra yellow unit squares to get to the following square.

enter image description here

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    $\begingroup$ Thank you, this is exactly the kind of intuition I was looking for. $\endgroup$ – jeremy radcliff May 7 '15 at 18:15
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    $\begingroup$ @jeremyradcliff There are many related illustrations on the web, see e.g. http://www.math.upenn.edu/~deturck/probsolv/LP1ans.html. $\endgroup$ – Jeppe Stig Nielsen May 8 '15 at 14:39
  • $\begingroup$ @JeppeStigNielsen, that's a great one too, thank you. $\endgroup$ – jeremy radcliff May 8 '15 at 16:10
  • $\begingroup$ ... and if the increment is $k$ (instead of $1$), the second increment in squares is $2k^2$ bigger than the first. $\endgroup$ – Joffan May 8 '15 at 19:22
  • $\begingroup$ IMHO even more intuition comes from shifting the yellow squares to the top right. $\endgroup$ – Mark Hurd May 12 '15 at 3:49
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We have the series $$ x_n = n^2 $$ and the differences $$ \Delta x_n = x_{n+1} - x_n = (n+1)^2 - n^2 = n^2 + 2n + 1 - n^2 = 2n + 1 $$ and the difference of differences $$ \Delta x_{n+1} - \Delta x_n = (2(n+1)+1) - (2n+1) = 2n + 3 - 2n - 1 = 2 $$

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The difference between two adjacent squares, $(a+1)^2$ and $a^2$:

$$\begin{align} (a+1)^2 - a^2 &= a^2 + 2a + 1 - a^2 \\ &= 2a + 1 \end{align}$$

So for each increase in $a$ by $1$, the delta between squares gets $2$ bigger.

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Hint: $(a+1)^2-a^2=a^2+2a+1-a^2=2a+1$

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Let $a$ be some natural number, and then let's think about $a+1$.
Since $a^2 = a \cdot a$, we have $(a+1)^2 = a^2 + 2a + 1$.

Then $(a+1)^2 - a^2 = 2a + 1$.

And then note for the numbers $1,2,3,\ldots$ applying $2a+1$ gives $2\cdot 1 + 1, 2 \cdot 2 + 1, 3 \cdot 2 + 1,\ldots$ Which is: $$3,5,7,9,\ldots$$

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