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I have this sum: $1+3+5+...+(2n+1)$, where $n$ is a natural number. I have to calculate it and interpret geometrically. Well, it's easy to find out that it equals $(n+1)^2$. But how to interpret it geometrically? I don't think it's about a graph(parabola). could you please give me at least an idea?

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  • $\begingroup$ That question doesn't address the geometric interpretation, though... $\endgroup$ – Johannes Kloos Nov 25 '13 at 21:22
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the sum of odd numbers is a square quilt

(Source: Wooly Thoughts afghans)

Or, more generally, try a search for "the sum of odd numbers is a square".

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Yes. You look for a square, also geometrically, with side $n+1$.

Slice it up for smaller squares, start with a unit square in a corner.

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Since this sum (n+1)^2 is the area of a square with side (n+1), draw a diagram of the n=1,2,3, ... squares with their lower left corner on the origin.

Enjoy!

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    $\begingroup$ There is also a nice configuration as a spiral from the centre $\endgroup$ – Mark Bennet Sep 18 '13 at 16:30