Why not




















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better indeed
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So, basically, 1+3+5+..+(2n-1) =(1+2+3+..+n)+(1+2+..n-1) =n(n+1)/2+n(n-1)/2 =(n+1+n-1)n/2 =(n/2)*2n =n^2
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I fail to see the first step you make :-?
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did something similar herehttps://beta.observablehq.com/@timhau/geometric-intuition-for-sum-of-first-n-odd-numbers …
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Alternately, a demonstration of how we derive that rule about squares from the behavior of physical objects.
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Well, those aren't exactly physical objects.
The Axioms were derivated from them, the formula derivates from the Axioms; no need to add it as another one 
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There are better visual proofs. Just look at the square tiles on your floor or ceiling.
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Similarly, Ulfilas of Milan (1508-1560) designed tiled walls for a small church near Fizzonasco. However he erred in the mathematical formula causing a grossly uneven pattern. The mistake was covered up with flocked wallpaper and cheesy lamps from Ross Dress for Less.
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