Sequence

A sequence is a set of numbers where the numbers are arranged in a definite order, like the natural numbers, the even integers between 10 and 100, or the squares of integers. In general, a sequence is written as $t_1, t_2, t_3, t_4, \ldots, t_n, \ldots$ where $t_1$ is the first term, $t_4$ is the fourth term, and so on up to $t_n$, the $n$th term.

A sequence containing a finite number of terms is called a finite sequence; it is written $\{t_1, t_2, t_3, \ldots, t_n\}$ for some positive integer $n$. A sequence that is not finite is called an infinite sequence; it is written $\{t_1, t_2, t_3, \ldots\}$ or $\{t_n\}_{n \ge 1}$.

Sequences that follow specific, recognisable patterns are called progressions. The previous class studied the Arithmetic Progression (A.P.); this chapter builds on that to study the Geometric Progression (G.P.), the Harmonic Progression (H.P.), the Arithmetico-Geometric Progression (A.G.P.), and finite/infinite series built from these.