Core answer: Two sequence families run the world: arithmetic (add a constant d: 3, 7, 11, 15…, sum = n(first+last)/2) and geometric (multiply by r: 3, 6, 12, 24…, sum = a(rⁿ−1)/(r−1)). Gauss's school trick — pairing 1+100, 2+99… — sums 1 to 100 as 50 × 101 = 5,050.
Arithmetic sequences (constant difference)
- nth term: aₙ = a₁ + (n−1)d
- Sum of n terms: Sₙ = n(a₁ + aₙ)/2 = n(2a₁ + (n−1)d)/2
Example — salary growth. Starting ¥8,000/month with a ¥500 annual raise: year-10 salary = 8,000 + 9×500 = ¥12,500. Total 10-year income = 10 × (8,000 + 12,500)/2 × 12 = ¥1,230,000.
Geometric sequences (constant ratio)
- nth term: aₙ = a₁ · rⁿ⁻¹
- Sum: Sₙ = a₁(rⁿ − 1)/(r − 1); infinite sum (|r| < 1): S∞ = a₁/(1 − r)
Example — compound interest IS a geometric sequence. ¥10,000 at 5%/year: after n years, 10,000 × 1.05ⁿ. The chessboard-rice legend (r = 2, 64 squares) totals 2⁶⁴ − 1 ≈ 1.8×10¹⁹ grains — more wheat than humanity has ever grown.
Example — infinite discount. A repeating decimal is a geometric series: 0.333… = 0.3/(1 − 0.1) = 1/3.
Famous sequences worth knowing
| Name | Start | Rule | Appears in |
|---|---|---|---|
| Fibonacci | 1, 1, 2, 3, 5, 8… | each = sum of previous two | phyllotaxis, story points |
| Square numbers | 1, 4, 9, 16… | n² | area, physics |
| Triangular | 1, 3, 6, 10… | n(n+1)/2 | handshakes, bowling pins |
| Powers of 2 | 1, 2, 4, 8… | ×2 | computing, epidemics |
Fibonacci ratios converge to φ = 1.618… (the golden ratio): 13/8 = 1.625, 21/13 = 1.615.
Common mistakes and myths
- Off-by-one on the exponent — the 1st term multiplies r⁰, so the nth term is a₁rⁿ⁻¹, not a₁rⁿ.
- Using the finite sum for |r| ≥ 1 infinite series — 1 + 2 + 4 + … diverges; infinite-sum formula requires |r| < 1.
- "Sequences must have a formula" — most real-world sequences (stock prices) have no closed form; pattern-spotting on 3 terms is guessing.
- Confusing arithmetic and geometric growth — linear +500/yr feels slow but beats 3% compounding for the first decade; always project both before investing in raises vs returns.
- Assuming Fibonacci explains every spiral — sunflowers yes (phyllotaxis), galaxies and nautilus shells only loosely; the golden-ratio myth oversells coincidence.