Sequences and series

Sequence vs series

A sequence is a list: a₁, a₂, a₃, … each position n has a term aₙ. A series is what you get when you add terms—often the partial sum of the first N terms. “How much after 12 deposits?” is a series question; “what is the 12th deposit?” is a sequence question.

Arithmetic: steady steps

Each term adds the same common difference d. Example: 3, 7, 11, 15 (d = 4). Formula:

aₙ = a₁ + (n − 1)·d

Real-life feel: a salary that rises by the same dollar amount each year; seats numbered down a row; a workout that adds 2 push-ups every session.

Geometric: steady multipliers

Each term multiplies by the same common ratio r. Example: 2, 6, 18, 54 (r = 3). Formula:

aₙ = a₁ · rⁿ⁻¹

Real-life feel: a photo scaled by 90% each time you compress; a chain letter style growth model; interest that multiplies a balance; half-life style decay with r = ½.