NapkinCalc

Algebra 2B — Exponentials, Logs & Sequences

Sequences & series

continues from lesson 2 — values defined earlier in the course stay live here

ELI5: adding 1 + 2 + ⋯ + n one at a time is a slog; pairing first with last is the insight — each pair sums to n + 1, and there are n/2 pairs, giving n(n + 1)/2. (Legend says Gauss found this at age 7.)

n=100n = 100
Sformula=n⋅(n+1)2S_{formula} = \frac{n \cdot \left(n + 1\right)}{2} = 50505050 Gauss's pairing trick: 5050
Sbrute:=Sum("i","i",1,100)S_{brute} := Sum("i", "i", 1, 100) = 50505050 the long way — all 100 additions
✓ pass Sformula==SbruteS_{formula} == S_{brute} the trick and the grind agree
Sgeo=1−0.5101−0.5S_{geo} = \frac{1 - 0.5^{10}}{1 - 0.5} = 1.99801.9980 geometric series 1 + ½ + ¼ + ⋯ (10 terms) — creeping up on 2

Real-world hook: geometric series price loans and annuities, sum a bouncing ball's total travel, and explain why 0.999… equals 1.

Try it yourself: use the formula n(n + 1)/2 to add 1 + 2 + ⋯ + 200, and let the engine check it against the brute-force sum.

sumyou=200⋅2012sum_{you} = \frac{200 \cdot 201}{2} = 2010020100 ✏️ Your turn: compute 1 + 2 + ⋯ + 200 with the formula n(n+1)/2 (n = 200). The check compares against grinding out all 200 additions.
✓ pass sumyou==Sum("i","i",1,200)sum_{you} == Sum("i", "i", 1, 200) green when your formula matches the brute-force sum

Where next: Trigonometry — where curves start to repeat and triangles take over.