Ryan Collison

Add the last two terms to get the next — and somehow sunflowers, shells, and stock charts follow the same rule. The Fibonacci sequence is arithmetic that keeps showing up in nature.

Each term multiplies by the same factor, and that single rule generates compound interest, radioactive decay, and viral spread. Geometric sequences are exponential growth hiding in plain arithmetic.

Arithmetic sequences are built from a single idea: add the same amount each step. That simplicity makes them predictable and powerful — from calculating compound savings to analyzing equally-spaced musical intervals.

The harmonic series — 1 + ½ + ⅓ + ¼ + ... — looks like it should converge. It doesn't. Understanding why is the first real lesson in the strangeness of infinity.

Can you add infinitely many numbers and get a finite result? Sometimes yes — and the conditions under which you can are exactly what series convergence is about. Sequences and series are where mathematics reasons rigorously about infinity, reaching into Taylor expansions and signal processing.

Mathematical intuition tells you something is true. Logic proves it must be. This synthesis shows how formal logic underpins all of mathematics — and why rigor is about structure, not difficulty.