Sequences — Introduction

A cricket team plays five matches in a tournament. After each match, you write down the team's total runs scored so far: 247, \; 491, \; 730, \; 982, \; 1231. That list has a definite order — the first number is the total after one match, the second after two, and so on. Rearranging the list would destroy its meaning. The order is the point.

Now consider something more abstract. Write down the perfect squares in order: 1, 4, 9, 16, 25, 36, \dots This list also has a rule and an order. The first entry is 1^2, the second is 2^2, the third is 3^2. You can compute any entry you want — the hundredth entry is 100^2 = 10{,}000 — without listing all the ones before it. The list goes on forever.

Both of these are sequences. The cricket example is finite (five terms, then the tournament ends). The perfect-squares example is infinite (there is no last perfect square). What they share is the essential property: an ordered list of numbers, produced by a rule, where the position of each number matters.

What a sequence really is

Strip away the examples and the word "list" and ask: what is a sequence, mathematically? It is a function whose domain is a set of consecutive positive integers.

Take the perfect-squares sequence. There is a function f that takes a positive integer n and returns n^2. The input is the position; the output is the value. f(1) = 1, f(2) = 4, f(3) = 9. The "list" 1, 4, 9, 16, \dots is just another way of writing the outputs of this function in order.

The notation a_n — with a subscript instead of parentheses — is standard. You will sometimes see (a_n)_{n \geq 1} when the author wants to be explicit about where the indexing starts.

The crucial point: order matters. The set \{1, 4, 9\} and the sequence 1, 4, 9 look similar, but the set does not care about order, while the sequence does. 4, 1, 9 is the same set as \{1, 4, 9\}, but it is a different sequence — its first term is 4, not 1.

Finite and infinite sequences

A finite sequence has a last term. The cricket example — 247, 491, 730, 982, 1231 — has exactly five terms. You can write it as (a_n)_{n=1}^{5} where a_1 = 247 and a_5 = 1231.

An infinite sequence has no last term. The perfect squares 1, 4, 9, 16, 25, \dots continue without end. No matter how far out you go — the millionth term, the billionth — there is always a next one. You write it as (a_n)_{n=1}^{\infty} or simply (a_n).

Most of the sequences you will meet in this series of articles — arithmetic progressions, geometric progressions, the Fibonacci sequence — are infinite. Finite sequences show up in practical situations: the scores of a fixed number of exams, the temperatures recorded on each day of a week, or the digits of a particular number.

The general term

The most common way to describe a sequence is with a general term (also called the n-th term formula). This is an expression in n that produces the n-th term when you plug in a specific value of n.

Take the sequence 2, 5, 8, 11, 14, \dots Each term is 3 more than the one before it. The first term is 2, the second is 2 + 3 = 5, the third is 2 + 2 \times 3 = 8. In general, the n-th term is

Check: a_1 = 3(1) - 1 = 2. a_2 = 3(2) - 1 = 5. a_5 = 3(5) - 1 = 14. The formula works.

The general term is powerful because it gives you random access. You do not need to compute a_1 through a_{99} to find a_{100} — you just plug in n = 100: a_{100} = 3(100) - 1 = 299. One step, done.

Here are a few more general terms to build your intuition:

Sequence	First few terms	General term a_n
Even numbers	2, 4, 6, 8, \dots	2n
Odd numbers	1, 3, 5, 7, \dots	2n - 1
Powers of 2	2, 4, 8, 16, \dots	2^n
Reciprocals	1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots	\frac{1}{n}
Alternating signs	1, -1, 1, -1, \dots	(-1)^{n+1}

Each formula takes the index n as input and returns the term as output. The formula is the sequence.

Drag the red point along the curve below to explore the sequence a_n = n^2. The readout shows the index and the term value — watch how the gap between consecutive terms grows as n increases.

Recursive definition

There is another way to describe a sequence. Instead of a formula that jumps straight to the n-th term, you give a starting value and a rule for computing the next term from the previous one.

Take the sequence 3, 7, 11, 15, 19, \dots You could give the general term a_n = 4n - 1. But you could also say:

This says: the first term is 3, and every term after that is obtained by adding 4 to the previous term. a_2 = a_1 + 4 = 7. a_3 = a_2 + 4 = 11. And so on.

This is a recursive definition — the word "recursive" meaning "defined in terms of itself." The sequence refers to its own earlier terms.

The trade-off between general-term and recursive definitions is access versus simplicity.

• A general term gives you random access. You can find a_{1000} instantly, without computing the 999 terms before it. But finding the general term can be hard.
• A recursive definition is often easier to spot. "Each term is 4 more than the last" is the kind of pattern you notice immediately. But to find a_{1000}, you would have to start at a_1 and apply the rule 999 times.

Some sequences have clean general terms. Others have clean recursive definitions. The Fibonacci sequence — 1, 1, 2, 3, 5, 8, 13, 21, \dots — has a beautifully simple recursive rule (a_{n+2} = a_n + a_{n+1}, each term is the sum of the two before it) but a much messier general term involving \sqrt{5}. On the other hand, the sequence 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots has a clean general term (a_n = \frac{1}{n}) but no particularly illuminating recursive rule.

Worked examples

Common confusions

• "A sequence must follow a simple pattern." Not necessarily. The sequence of prime numbers — 2, 3, 5, 7, 11, 13, 17, \dots — is a perfectly valid sequence, but there is no simple formula that produces the n-th prime from n. A sequence only needs a definite rule assigning a value to each index; the rule does not have to be a tidy algebraic expression.

• "The dots '\dots' mean I can guess what comes next." The dots mean the sequence continues according to the stated rule. Without a rule, you cannot know what comes next. The terms 1, 2, 4, \dots could continue as 8, 16, 32, \dots (powers of 2) or as 7, 11, 16, \dots (add 1, add 2, add 3, ...) or infinitely many other ways. The general term or recursive definition removes the ambiguity.

• "A sequence and a set are the same thing." They are not. The set \{1, 4, 9\} and the sequence 9, 4, 1 contain the same numbers, but the sequence has an order and can have repeated entries, while the set has neither. The sequence 1, 1, 1, 1, \dots has infinitely many terms; the set of its values is just \{1\}.

• "To find a_{100}, I must compute all previous terms." Only if you are using a recursive definition. With a general term, you plug in n = 100 directly. This is exactly the advantage of having a general term.

• "Every recursive sequence needs only one starting value." Some recursive definitions look back more than one step. The Fibonacci sequence needs two starting values (a_1 = 1 and a_2 = 1) because the rule a_{n+2} = a_n + a_{n+1} refers to two previous terms. In general, a rule that looks back k steps needs k starting values.

Going deeper

Sequences as functions on \mathbb{N}

Formally, an infinite sequence of real numbers is a function a : \mathbb{N} \to \mathbb{R}, where \mathbb{N} = \{1, 2, 3, \dots\}. The domain is the set of positive integers and the codomain is the real numbers. When you write a_n = n^2, you are defining this function by a formula.

A finite sequence of length m is a function a : \{1, 2, \dots, m\} \to \mathbb{R}. The only difference is the domain.

This function viewpoint explains why order matters (the function assigns different values to different inputs) and why repetition is allowed (the function can take the same value at multiple inputs — the constant sequence a_n = 5 has every term equal to 5, which is fine for a function even though a set would collapse it to one element).

Recursive definitions and induction

There is a deep link between recursive definitions and mathematical induction. When you define a sequence recursively — say a_1 = 3, a_{n+1} = a_n + 4 — how do you know that every term is well-defined? You know a_1 exists (it is given). And if a_n exists, then a_{n+1} = a_n + 4 exists too. By induction, a_n exists for every positive integer n.

This is not just an analogy; it is the precise mathematical justification. The recursion theorem (a consequence of the axioms of the natural numbers) guarantees that a recursive definition with a valid base case and a valid recursive step produces a unique well-defined sequence. The proof of the recursion theorem uses induction.

So every time you write a recursive definition and trust that it defines a sequence, you are implicitly relying on induction to guarantee that the construction works at every step.

Monotonic and bounded sequences

Two properties of sequences that matter greatly when you study convergence later:

A sequence (a_n) is monotonically increasing if a_{n+1} \geq a_n for all n, and monotonically decreasing if a_{n+1} \leq a_n for all n. The sequence 1, 4, 9, 16, \dots is increasing. The sequence 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots is decreasing.

A sequence is bounded above if there is a number M such that a_n \leq M for all n, and bounded below if there is a number m such that a_n \geq m for all n. If it is bounded both above and below, it is simply called bounded.

The sequence \frac{1}{n} is bounded (it stays between 0 and 1) and monotonically decreasing. The sequence n^2 is bounded below (by 1) but not bounded above — it grows without limit.

A central theorem you will meet in calculus: every bounded, monotonically increasing sequence converges to a limit. That theorem is the reason these two properties matter, and it is one of the pillars of analysis.

Where this leads next

Sequences are the starting point for a family of closely related ideas.

• Arithmetic Progression — the sequence where each term is a fixed amount more than the previous one. The general term, the sum, and the properties that make APs the most common sequences in school mathematics.
• Geometric Progression — the sequence where each term is a fixed multiple of the previous one. Growth, decay, compound interest, and infinite sums.
• Mathematical Induction — the proof technique that lets you prove a statement about a sequence (or any property of positive integers) by proving it for one index and then for the step from n to n+1.
• Sets — Introduction — the language of sets underpins the formal definition of a sequence as a function from a subset of the integers to the reals.
• Algebraic Expressions — the formulas you write for general terms are algebraic expressions in n, and the skills from that topic — simplifying, factoring, substituting — are exactly what you need to work with sequences.