Determining the General Term $T_n$ of the Sequence 18, 16, 12, 6, -2, ...

Overview

This animation discovers the explicit formula for the nth term $T_n$ of the given decreasing sequence $18, 16, 12, 6, -2, \dots$. Viewers see the pattern of first and second differences, recognize a quadratic relationship, and obtain the closed‑form expression.

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Phases

#	Phase Name	Duration	Description
1	Intro	~3s	Present the initial terms of the sequence on the screen, fading in one by one.
2	Difference Analysis	~8s	Compute and display first differences, then second differences, highlighting the constant second difference.
3	Quadratic Model Derivation	~10s	Show the generic quadratic form $T_n = an^2 + bn + c$, substitute three consecutive terms, solve for $a, b, c$ step‑by‑step.
4	Final Formula Reveal	~4s	Present the derived formula $T_n = -n^2 + 5n + 12$ and verify it against the original terms.
5	Outro	~2s	Summarize the key takeaway: constant second difference ⇒ quadratic term.

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Layout

┌─────────────────────────────────────────────┐
│                                             │
│                MAIN (visual)                │
│   Sequence, differences, algebraic steps     │
│                                             │
├─────────────────────────────────────────────┤
│  Caption (optional) – current step title    │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Main	Numbers of the sequence, tables of first/second differences, algebraic symbols, and the final formula.	All visual; no static text blocks except necessary symbols.
Caption	Short label indicating the current phase (e.g., "First Differences").	Appears only when helpful; otherwise omitted.

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Notes

• Assumptions: The sequence follows a quadratic pattern because the second differences become constant (‑2). The animation stays within a 30‑second limit.
• No persistent explanatory text is used; all information is conveyed through numbers, arrows, and highlighted algebraic steps.
• Transitions are simple fades and slide‑ins to keep the pacing brisk.
• The final verification shows the first five terms generated by the formula matching the original list.
• The scene will be implemented as a single Manim Scene class.