Sum of Consecutive Odd Numbers Gives Perfect Squares

Overview

A short visual proof that adding the first n odd numbers yields the perfect square $n^2$. The animation builds the sum step‑by‑step, shows how each new odd term adds a layer around a growing square, and ends with the general formula.

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Phases

#	Phase Name	Duration	Description
1	Intro	~3s	Title fades in: “Why $1+3+5+\dots+(2n-1)=n^2$”. A single dot representing $n=1$ appears as a 1×1 square.
2	Adding Odd Terms	~10s	For $k=1$ to $4$ (illustrative), each odd number $2k-1$ is shown as a row of unit squares that attach to the right side and then a column that attach to the bottom, forming a larger square. The cumulative sum is highlighted.
3	General Pattern	~5s	A schematic diagram appears: a growing $n\times n$ grid with a highlighted L‑shaped layer labeled $2n-1$. The equation $1+3+5+\dots+(2n-1)=n^2$ is displayed briefly.
4	Conclusion	~4s	The final $n\times n$ square expands to fill the screen, while the formula $\displaystyle \sum_{k=1}^{n}(2k-1)=n^2$ stays centered. Fade out.

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Layout

┌─────────────────────────────────────────────┐
│                                             │
│                MAIN (visual proof)          │
│                                             │
├─────────────────────────────────────────────┤
│  Caption (optional, small) – formula overlay │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Main	Unit squares being added, forming larger squares; L‑shaped layer illustrating each odd term.	All visual steps occur here.
Caption	Brief formula overlay (e.g., $\sum_{k=1}^{n}(2k-1)=n^2$).	Appears only in Phases 3‑4, small font at bottom.

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Notes

• Keep the animation under 25 seconds; timings above are approximate and can be trimmed.
• Use a consistent color scheme: base squares in light gray, newly added layer in a contrasting hue (e.g., teal).
• No textual narration is required; the visual progression should be self‑explanatory.
• The proof is demonstrated for a few concrete values (1‑4) before abstracting to the general case.
• Ensure the final formula stays on screen for at least 2 seconds to allow the viewer to read it.