interpret a sum geometrically

Question

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?

That question doesn't address the geometric interpretation, though... — Johannes Kloos, Nov 25 '13 at 21:22

score 8 · Accepted Answer · answered Sep 18 '13 at 16:22

8

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".

answered Sep 18 '13 at 16:22

MJD

65,394
39
298
580

score 1 · Answer 2 · answered Sep 18 '13 at 16:13

1

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.

answered Sep 18 '13 at 16:13

Berci

90,745

score 1 · Answer 3 · answered Sep 18 '13 at 16:19

1

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!

answered Sep 18 '13 at 16:19

Anton1701

21

1

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

interpret a sum geometrically

3 Answers3