Uploaded September 2026 | Updated September 2026, 2 weeks ago

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Summing odds via area

This is a short, animated visual proof showing one method to find the sum of the first n odd positive integers decomposing an array of unit squares into rectangular and triangular pieces and using area formulas for those pieces.

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To see the other methods for finding the sum odd integers visually, check out the following videos:
youtu.be/jq5AYCgkciE?si=U6bxLBmWjyBD_HQZ
youtu.be/I4A2T55k0cg
youtu.be/x3qfFBNRRDg
youtu.be/ZeEOgbLo0Rg
youtu.be/6H6nCpGW1nU

For other sums related to odd integers, check out these visuals:
youtu.be/WKxoi2i0EaE (summing odd squares)
studio.youtube.com/video/9BC1klE8jMk/edit (alternating odd sums)
youtu.be/MmOTqrtbtFQ (odd sum ratios)
youtu.be/fTBvVeURb3Q (odd sum ratios II)

This animation is based on a visual proof due to Samuel G. Moreno that appeared in the October 2017 issue of Mathematics Magazine (doi.org/10.4169/math.mag.90.4.298 page 298).

#manim #math #proofwithoutwords #visualproof #squarenumbers #oddnumbers

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https://manim.community