Break What? Forum

What is High School / Foundations

a Sequence?

An ordered list of numbers following a rule — the foundation of series, limits, and all of infinite mathematics.

Simple Definition

A sequence is an ordered list of numbers generated by a rule. Each number in the list is called a term. The first term is \(a_1\), the second is \(a_2\), and so on. The \(n\)-th term is \(a_n\).

The key word is ordered — position matters. The sequence \(1, 2, 3\) is different from \(3, 2, 1\). Unlike a set (which has no order and no duplicates), a sequence can repeat values and the order is essential.

Sequences can be finite (a fixed number of terms) or infinite (going on forever). Infinite sequences are the ones that connect to limits, convergence, and series in calculus.

Intuition

🎵

A playlist with a pattern

Think of a sequence like a playlist where each song is chosen by a rule: "always play a song 3 BPM faster than the last." Start at 100 BPM → 103 → 106 → 109... That's an arithmetic sequence. The rule generates every term — you don't need to list them all, just know the rule.

🔭

Does it settle down?

The most important question about an infinite sequence: does it approach a fixed value as \(n \to \infty\)? The sequence \(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots\) approaches 0 — it converges. The sequence \(1, 2, 3, 4, \ldots\) grows without bound — it diverges. Convergence is the bridge between sequences and limits.

Types of Sequences

Arithmetic

Add a constant each step

\(a_n = a_1 + (n-1)d\)

2, 5, 8, 11, 14, … (d = 3)

Geometric

Multiply by a constant each step

\(a_n = a_1 \cdot r^{n-1}\)

3, 6, 12, 24, 48, … (r = 2)

Recursive

Each term defined by previous terms

\(a_n = a_{n-1} + a_{n-2}\)

1, 1, 2, 3, 5, 8, 13, … (Fibonacci)

Explicit

Formula directly gives \(a_n\)

\(a_n = \dfrac{1}{n}\)

1, ½, ⅓, ¼, ⅕, … → converges to 0

Visual — Sequences & Convergence

Arithmetic: a_n = 2 + 3(n−1) — grows linearly without bound

Key Formulas

Arithmetic nth term	\(a_n = a_1 + (n-1)d\)
Arithmetic sum (first n)	\(S_n = \dfrac{n}{2}(a_1 + a_n)\)
Geometric nth term	\(a_n = a_1 \cdot r^{n-1}\)
Geometric sum (first n)	\(S_n = a_1\dfrac{1-r^n}{1-r}\quad(r \neq 1)\)
Convergence condition (geo)	Converges if and only if \(\|r\| < 1\)
Limit of a sequence	\(\lim_{n\to\infty} a_n = L\) means terms approach \(L\)

Quick Examples

Example 1 — Arithmetic nth term

First term 4, common diff 7

\(a_n = 4 + (n-1) \cdot 7 = 7n - 3\)
\(a_{10} = 70 - 3 = 67\)

Example 2 — Geometric

\(a_n = 5 \cdot (1/2)^{n-1}\)

5, 2.5, 1.25, 0.625, ...
Converges to 0 since \(|r| = 1/2 < 1\)

Example 3 — Limit

\(a_n = \dfrac{3n+1}{n}\)

\(= 3 + \dfrac{1}{n} \to 3\) as \(n \to \infty\)
Sequence converges to 3

Example 4 — Diverges

\(a_n = (-1)^n\)

−1, 1, −1, 1, −1, …
Oscillates forever — no limit, diverges

Common Mistakes

✗ Confusing sequence and series

A sequence is a list: \(1, 2, 3, 4, \ldots\). A series is a sum: \(1 + 2 + 3 + 4 + \ldots\). The sequence is the raw terms; the series is what you get when you add them. These are completely different objects with different convergence behaviour.

✗ Assuming a bounded sequence converges

A bounded sequence (all terms stay between two values) does not necessarily converge. The sequence \((-1)^n\) is bounded between −1 and 1 but oscillates forever and diverges. You also need the terms to be monotone (always increasing or always decreasing) to guarantee convergence.

✗ Using n = 0 vs n = 1 without care

Sequences can be indexed starting from \(n = 0\) or \(n = 1\). The formula \(a_n = a_1 \cdot r^{n-1}\) assumes \(n\) starts at 1. If \(n\) starts at 0, use \(a_n = a_0 \cdot r^n\). Always check which convention your formula uses.

Related Concepts

Ready to test your knowledge?

Sequences appear in High School, Calculus II, and beyond on BUders.

Browse All Quizzes →