Numerical Sequences: Ways to Define Sequences
1. What is a Numerical Sequence?
Definition: A numerical sequence is an ordered list of numbers that follows a specific rule or pattern.
Think of a sequence as numbers lined up in a specific order, where each number has its own position:
$$ a_1, \; a_2, \; a_3, \; \ldots, \; a_n, \; \ldots $$
Key Terminology
| Term | Meaning | Example |
|---|---|---|
| Term | Each individual number in the sequence | In $2, 5, 8, 11, \ldots$ the first term is $2$ |
| Index (subscript) | The position number $n$ of a term | $a_3$ means the term at position $3$ |
| General term ($a_n$) | A formula representing any term at position $n$ | $a_n = 3n - 1$ gives $a_1=2, a_2=5, a_3=8$ |
Infinite Sequence
Continues indefinitely — has no last term.
$a_1, a_2, a_3, \ldots, a_n, \ldots$
Example: $1, 4, 9, 16, 25, \ldots$
Finite Sequence
Has a fixed number of terms — stops at some $a_n$.
$a_1, a_2, \ldots, a_n$
Example: $2, 4, 6, 8, 10$ (5 terms)
Sequence as a Function: An infinite sequence can be viewed as a function $f: \mathbb{N} \to \mathbb{R}$, where $f(n) = a_n$. Each natural number $n$ maps to exactly one real number $a_n$.
2. Three Ways to Define a Sequence
Way 1: Explicit Formula (n-th Term Formula)
An explicit formula lets you calculate any term directly from its position number $n$, without needing to know previous terms.
Example: $a_n = 2^n + (-1)^n \cdot n$
- $a_1 = 2^1 + (-1)^1 \cdot 1 = 2 - 1 = \mathbf{1}$
- $a_2 = 2^2 + (-1)^2 \cdot 2 = 4 + 2 = \mathbf{6}$
- $a_7 = 2^7 + (-1)^7 \cdot 7 = 128 - 7 = \mathbf{121}$
- $a_{10} = 2^{10} + (-1)^{10} \cdot 10 = 1024 + 10 = \mathbf{1034}$