Sequences and Series Summary

Algebra 2 · Axiom Academy

From discrete patterns to cumulative sums — how sequences become series, and when an infinite sum actually converges. Every sequence is either arithmetic (constant difference d ) or geometric (constant ratio r ) — spotting which one tells you exactly which formula to reach for. A sequence can be written explicitly ( a_n as a direct function of n ) or recursively (each term built from the one before it) — the same pattern, two different lenses. A series is just a sum of a sequence's terms. Sigma notation ( ) is the shorthand for it, and it's linear — a sum of combined terms always splits apart. Finite sums have closed-form shortcuts (Gauss's pairing trick for arithmetic, telescoping for geometric); an infinite geometric series only sums to a finite number when . These aren't abstract exercises — compound interest, population growth, and depreciation are all geometric sequences in disguise. Core Concept Sequences: Explicit vs. Recursive A sequence is an ordered list of numbers — formally, a function whose domain is the positive integers. The same pattern can always be described two ways. Explicit: a_n written directly as a function of n — jump straight to any term without finding the ones before it. Recursive: each term built from the term(s) before it, plus a starting value — natural for "what changes each step" patterns like Fibonacci. Core Concept Arithmetic Sequences

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