Sequence

Calculus
definition

Also known as: number sequence, sequences

Grade 9-12

View on concept map

An ordered list of numbers generated by a rule, where each number has a specific position (first, second, third, . Sequences are the foundation for series, limits, convergence, and patterns in higher mathematics.

This concept is covered in depth in our understanding number sequences and patterns, with worked examples, practice problems, and common mistakes.

Definition

An ordered list of numbers generated by a rule, where each number has a specific position (first, second, third, ...).

💡 Intuition

A pattern of numbers: first term, second term, third term, and so on.

🎯 Core Idea

A sequence is an ordered list of numbers defined by a rule; each term has a specific position (index).

Example

1, 4, 9, 16, 25... (squares). 1, 1, 2, 3, 5, 8... (Fibonacci).

Formula

\{a_n\}_{n=1}^{\infty} = a_1, a_2, a_3, \ldots \quad \text{Converges if } \lim_{n \to \infty} a_n = L

Notation

a_n = nth term

🌟 Why It Matters

Sequences are the foundation for series, limits, convergence, and patterns in higher mathematics.

💭 Hint When Stuck

Write out the first 5 or 6 terms explicitly to spot the pattern before trying to find a formula.

Formal View

A sequence is a function a : \mathbb{N} \to \mathbb{R}, written (a_n)_{n=1}^{\infty}. The sequence converges to L if \forall \epsilon > 0,\; \exists N \in \mathbb{N} : n > N \implies |a_n - L| < \epsilon.

🚧 Common Stuck Point

A sequence can converge (approach a limit) or diverge (grow without bound).

⚠️ Common Mistakes

  • Confusing a sequence with a series: a sequence is a list of numbers (a_1, a_2, a_3, \ldots), while a series is the sum of those numbers (a_1 + a_2 + a_3 + \ldots).
  • Writing the general term formula with wrong indexing: if the sequence starts at n = 1 with value 3, the formula a_n = 2n + 1 gives a_1 = 3 but a_n = 2n gives a_1 = 2 — always verify with the first few terms.
  • Assuming a sequence that increases must diverge: the sequence a_n = 1 - \frac{1}{n} is increasing but converges to 1.

Frequently Asked Questions

What is Sequence in Math?

An ordered list of numbers generated by a rule, where each number has a specific position (first, second, third, ...).

Why is Sequence important?

Sequences are the foundation for series, limits, convergence, and patterns in higher mathematics.

What do students usually get wrong about Sequence?

A sequence can converge (approach a limit) or diverge (grow without bound).

How Sequence Connects to Other Ideas

Once you have a solid grasp of sequence, you can move on to arithmetic sequence, geometric sequence and series.

Want the Full Guide?

This concept is explained step by step in our complete guide:

Growing Patterns, Arithmetic and Geometric Sequences →
← Back to Explorer

Visualization

Static

Visual representation of Sequence