Arithmetic and Geometric Sequences

Nature behaves in both continuous sweeps and discrete jumps. When a physicist models the trajectory of a thrown ball, they use a continuous curve because the ball occupies every infinitesimally small point in space along its path. However, when a school board allocates funding based on yearly enrollment, or a bank computes interest at the end of a month, the timeline fractures into distinct, countable steps. In mathematics, a sequence is a mathematical function whose domain is a subset of integers. More specifically, to align with the act of counting events or steps, the domain of a mathematical sequence is typically restricted to positive integers.

When we introduce sequence notation, the variable $n$ is not merely an arbitrary placeholder like $x$ in a continuous equation; the variable $n$ in sequence formulas represents the ordinal position of the term—the first term, the second term, the third, and so on. Because it denotes a discrete position, the sequence position variable $n$ must be an integer. As a secondary mathematics teacher, your task is to help students bridge their understanding of continuous functions (lines and curves drawn without lifting the pencil) to these discrete, stepwise realities.