A sequence is an ordered, never-ending list of numbers:
a_1,\ a_2,\ a_3,\ a_4,\ \dots
The clean way to say it: a sequence is just a
function
whose inputs are the natural numbers. Feed it a position
n \in \{1, 2, 3, \dots\} and it hands back the
n-th term,
a_n:
a : \mathbb{N} \to \mathbb{R}, \qquad n \mapsto a_n.
We write the whole sequence as (a_n)_{n \ge 1}, or just
(a_n). The subscript is the input; the term is the output. So
"the sequence of squares" 1, 4, 9, 16, \dots is the rule
a_n = n^2 — a single formula, the closed form,
that produces any term you ask for without working up to it.
Spot the pattern, write the closed form
Most of the work of a first course on sequences is reading a list and recovering its rule.
Here are the four patterns you will meet again and again. In each, watch the same move:
stare at consecutive terms, find what stays constant, and turn it into a formula for
a_n.
Step 1 — Arithmetic: a constant difference. Take
3, 7, 11, 15, \dots Each term is 4 more
than the last, so the gap d = a_{n+1} - a_n = 4 never changes.
Starting at a_1 = 3 and adding 4 a total
of n-1 times:
a_n = a_1 + (n-1)\,d = 3 + 4(n-1) = 4n - 1.
Step 2 — Geometric: a constant ratio. Take
2, 6, 18, 54, \dots Now each term is 3
times the last, so the ratio r = a_{n+1}/a_n = 3 is fixed.
Starting at a_1 = 2 and multiplying by 3
a total of n-1 times:
a_n = a_1\, r^{\,n-1} = 2 \cdot 3^{\,n-1}.
Step 3 — Harmonic: the reciprocals of the counting numbers. The list
1, \tfrac12, \tfrac13, \tfrac14, \dots has neither a constant
difference nor a constant ratio, but its closed form is the simplest of all:
a_n = \frac{1}{n}.
Its terms shrink toward 0 — a fact we will pin down exactly when
we define the limit of a sequence.
Step 4 — Recursive: a term defined from earlier terms. Sometimes the rule
is not a closed form at all, but a recurrence — a starting value plus an instruction
for the next term. The Fibonacci sequence is the classic:
a_1 = 1,\quad a_2 = 1,\qquad a_{n} = a_{n-1} + a_{n-2}\ \ (n \ge 3),
giving 1, 1, 2, 3, 5, 8, 13, \dots A recurrence always tells you
how to grind out the next term; whether a tidy closed form also exists is a separate,
often harder, question.
A sequence of real numbers is a function
a : \mathbb{N} \to \mathbb{R}, \qquad a(n) = a_n,
written (a_n)_{n \ge 1}. It may be specified two ways, which are
interchangeable when both exist:
-
Closed form — a direct formula for the
n-th term, e.g. a_n = 4n - 1; ask
for any term and get it in one step.
-
Recurrence — initial term(s) plus a rule producing each term from its
predecessors, e.g. a_n = a_{n-1} + a_{n-2}; you must build up
to the term you want.
Nothing forces a sequence to start at n = 1. Power series and
computer scientists love starting at n = 0, so
(a_n)_{n \ge 0} runs a_0, a_1, a_2, \dots
Either convention is fine — it only shifts the bookkeeping. A geometric sequence indexed
from 0 reads a_n = a_0\, r^{\,n} (no
n-1), which is why the
geometric series
is usually written \sum_{n \ge 0} a r^n. Pick a convention,
state it, and stay consistent.