NapkinCalc

Algebra 2

Sequences & series

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

Adding 1 + 2 + ⋯ + n one at a time is work; pairing first with last is insight — every pair sums to n + 1, and there are n/2 pairs. The check makes the engine grind out the long way and compares:

n=100n = 100
Sformula=n⋅(n+1)2S_{formula} = \frac{n \cdot \left(n + 1\right)}{2} = 50505050 Gauss's pairing trick
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

Where next: Trigonometry — where the curves start repeating and triangles take over.