What is a number sequence?

It is a function whose domain is the natural numbers. It assigns exactly one number to every index n, called the nth term and written aₙ rather than a(n). The whole difference from an ordinary function lies in the domain: the argument may be 1, 2 or 3, but never 2.5.

Why is the graph of a sequence made of dots rather than a curve?

Because the domain is the natural numbers, and there is no natural number between 3 and 4. So no term exists that could be drawn between the third and the fourth dot. Joining the dots with a line would suggest values the sequence simply does not have.

How does a general formula differ from a recursive one?

A general formula gives a term straight from its index: substitute n = 100 and the hundredth term is one calculation away. A recursive formula gives a recipe for the next term from the previous one, so reaching the hundredth means computing all ninety-nine before it. Both can describe the same sequence.

How do I check whether a sequence is increasing?

Compute the difference of consecutive terms aₙ₊₁ − aₙ and look at its sign. Positive for every n means increasing; negative for every n means decreasing. Checking the first few terms proves nothing, because the sign of the difference can change further along.

Is every sequence either increasing or decreasing?

No. The sequence aₙ = n² − 6n has difference 2n − 5, which is negative for n = 1 and n = 2 and positive from n = 3 on — such a sequence falls first and rises afterwards, so it is not monotonic. The sequence aₙ = (−1)ⁿ jumps between −1 and 1 and is neither increasing nor decreasing either.

Which index does a sequence start at?

Most often n = 1, and that is the convention used here: the first term is a₁. Sequences indexed from zero also occur — then a₀ is the first term. The indexing is a matter of notation, but it has to be stated, because it changes the value of every term computed from the general formula.

Can a sequence have finitely many terms?

Yes — a finite sequence has its domain cut down to the indices from 1 to k. A list of twelve monthly meter readings looks like that. An infinite sequence has all the natural numbers as its domain, and only for such a sequence does the question of a limit make sense.

How do I write down a sequence given as a list of terms?

You have to find a rule and then check it against every term given. For the list 1, 4, 9, 16 the rule is aₙ = n²; for 2, 4, 8, 16 it is aₙ = 2ⁿ. A list alone never determines the formula uniquely, which is why an exercise states the formula rather than the first few numbers.

Intermediate level

Number sequences

A sequence is a function that assigns a number to every natural number — which is exactly why its graph is a set of separate dots rather than a curve. See how a general formula works, how a recursive one differs from it, and how a single subtraction settles whether a sequence increases.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

What a function isA function is an unambiguous assignment: every argument gets exactly one value. Meet the f(x) notation, the domain and the range, the four ways to define a function, and the vertical line test on a graph.
Number setsNatural, integer, rational and real numbers — four families, each one contained in the next. Learn the symbols ℕ, ℤ, ℚ, ℝ, the membership signs ∈ and ∉, the empty set ∅, the proof that √2 is irrational, and the 2k / 2k+1 notation every divisibility proof is built on.

Where this is used

Real situations where you count exactly the way this lesson teaches:

Training plan
A ten-week running plan: 20 minutes of easy running in week one, 5 minutes more in every week after that. That is a sequence with the general term aₙ = 15 + 5n, so week ten calls for 15 + 50 = 65 minutes. The difference between consecutive terms is a constant 5 minutes — positive, so the load rises and no week is lighter than the one before it.
Convert it here: Time
Handshakes at a meeting
n people arrive at a meeting and everyone shakes hands with everyone else. The number of handshakes is the sequence aₙ = n(n − 1)/2: for 8 people that is 8 · 7 / 2 = 28, and for 12 people already 66. The increase is not constant — the ninth person adds 8 handshakes and the twelfth adds 11, so this sequence grows faster and faster.
Drug dosing
A drug taken once a day: after 24 hours 40% of the previous dose is still in the body, and the patient takes another 100 mg. That is the recursive formula aₙ₊₁ = 0.4 · aₙ + 100 with a₁ = 100 mg. The next values are 100, 140, 156 and 162.4 mg — the concentration rises, ever more slowly, and never passes 166.7 mg. A pharmacologist computes exactly this sequence to check that the saturation level sits inside the therapeutic window.
Depreciating equipment
An accountant books a laptop worth 6000 units, depreciated in a straight line over five years, so 1200 units a year. The book value after n years is the sequence aₙ = 6000 − 1200n: 2400 after three years, zero after five. The difference between consecutive terms is −1200, negative for every n, so the sequence is decreasing — which matches the fact that equipment never gains book value.

All formulas

A sequence is a function
$a_n = f(n), \qquad n \in \mathbb{N}_+$
the domain is the natural numbers, so the graph is a set of dots
General term
$a_n = 2n + 1$
substitute the index and the term is there
Recursive formula
$a_1 = 3, \qquad a_{n+1} = 2a_n - 1$
each term from the previous one — you have to walk through them
Increasing sequence
$a_{n+1} - a_n > 0$
the difference of consecutive terms is positive for every n
Decreasing sequence
$a_{n+1} - a_n < 0$
the same difference, negative for every n
Constant sequence
$a_{n+1} - a_n = 0$
every term equal

The notation $a_1, a_2, a_3, \ldots$ looks like an ordinary list of numbers, but something familiar hides behind it: a function. One with an unusually thrifty domain.

A sequence is a function on the natural numbers

A sequence assigns exactly one number to every natural number. That is the definition of a function — the only thing that changes is what may go in as the argument:

a_n = f(n), \qquad n \in \mathbb{N}_+

Instead of $a(n)$ we write $a_n$ , and the argument is called the index of the term. The number $a_n$ is the $n$ th term of the sequence; $a_1$ is the first one.

The domain holds only natural numbers, and those are separated from one another: there is nothing between $3$ and $4$ . That is why the graph of a sequence consists of separate dots.

The graph of a(n) = 2n + 1. The dots do lie on one straight line, but we never draw that line — the sequence has no term numbered 2.5.

The dots line up along a straight line here because the formula is linear. That does not mean the graph is a line: the value $a_{2.5}$ does not exist, so there is nothing to draw between the second dot and the third.

The general term

A general formula gives a term straight from its index. Substituting is all it takes:

a_n = n^2 + 1

a_1 = 1^2 + 1 = 2, \quad a_2 = 2^2 + 1 = 5, \quad a_3 = 3^2 + 1 = 10, \quad a_4 = 17

The great advantage of this notation: every term is one calculation away. The hundredth is $a_{100} = 100^2 + 1 = 10\,001$ — and there is no need to walk through the ninety-nine before it.

A sequence is given by aₙ = 3n² − 4n. Find a₅ and a₁ + a₂.

The recursive formula

The second way to describe a sequence does not say what $a_n$ is; it says how to make the next term out of one you already have:

a_1 = 3, \qquad a_{n+1} = 2a_n - 1

Such a recipe is called a recursive formula, and it always has two parts: a starting term and a step rule. Without the first part the sequence is not determined — the rule on its own does not know where to begin.

The terms come out one after another:

a_2 = 2 \cdot 3 - 1 = 5, \qquad a_3 = 2 \cdot 5 - 1 = 9, \qquad a_4 = 2 \cdot 9 - 1 = 17

$n$	$1$	$2$	$3$	$4$	$5$
$a_n$	$3$	$5$	$9$	$17$	$33$

And here is the whole difference from a general formula: there is no shortcut to $a_{100}$ . Every earlier term has to be computed, one by one. In exchange, recursion describes beautifully the processes where the next state really does come out of the previous one — an account balance after interest is added, a drug concentration after another dose, a population after another season.

A sequence is given by a₁ = 2, aₙ₊₁ = 3aₙ + 1. Find a₄.

Monotonicity: one subtraction

A sequence is increasing when every term is greater than the one before it. Rather than staring at the numbers, look at the sign of the difference:

a_{n+1} - a_n

the difference is positive for every $n$ — an increasing sequence;
the difference is negative for every $n$ — a decreasing sequence;
the difference equals zero — a constant sequence.

For $a_n = 2n + 1$ one calculation answers the question for all terms at once:

a_{n+1} - a_n = \bigl(2(n+1) + 1\bigr) - (2n + 1) = 2n + 3 - 2n - 1 = 2 > 0

The difference came out constant and positive, so the sequence increases — always, not merely at the start.

The decreasing sequence b(n) = 10 − 2n. The difference of consecutive terms is −2, so the dots step down by the same amount every time.

The trap most readers fall into: checking the first few terms is not a proof. Take $a_n = n^2 - 6n$ :

a_{n+1} - a_n = \bigl((n+1)^2 - 6(n+1)\bigr) - (n^2 - 6n) = 2n - 5

For $n = 1$ and $n = 2$ the difference is negative; from $n = 3$ on it is positive. The sequence falls first ( $-5, -8, -9$ ) and rises afterwards ( $-8, -5, 0$ ) — it is not monotonic, however strongly the first three terms suggested otherwise.

Determine whether the sequence aₙ = 7 − 4n is monotonic.

Sequences with a constant difference between consecutive terms — like $2n + 1$ or $7 - 4n$ — have a name of their own and formulas of their own for the sum. That is the material of the next lesson, on arithmetic and geometric sequences.

Exercises

The exercises drill the two procedures of this lesson: reading a term off the general formula (substitute the index) and getting one out of a recursion (walk forward from the first term). The answer is a whole number, negative ones included. Monotonicity is not asked here — its answer is a word rather than a number, so you practise it on the examples above.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

aₙ = 3n; a₅ =

Common mistakes

Joining the dots of the graph with a line — a sequence has no terms at fractional indices, so there is nothing to draw between the dots.
Substituting the index into a recursive formula — in $a_{n+1} = 2a_n - 1$ the slot $a_n$ takes the previous term, not the number $n$ .
A recursion without a starting term — the step rule alone does not determine a sequence; $a_1$ is part of the formula.
Calling a sequence increasing after looking at three terms — what settles it is the sign of $a_{n+1} - a_n$ for every $n$ .
Confusing the index with the value — $a_5 = -12$ is perfectly fine: the index has to be natural, the term does not.
An off-by-one in the indexing — if a sequence starts at $a_1$ , then $a_1$ is the first term, not the zeroth.

Formula card

Topic: Number sequences

A sequence is a function
$a_n = f(n), \qquad n \in \mathbb{N}_+$
the domain is the natural numbers, so the graph is a set of dots
General term
$a_n = 2n + 1$
substitute the index and the term is there
Recursive formula
$a_1 = 3, \qquad a_{n+1} = 2a_n - 1$
each term from the previous one — you have to walk through them
Increasing sequence
$a_{n+1} - a_n > 0$
the difference of consecutive terms is positive for every n
Decreasing sequence
$a_{n+1} - a_n < 0$
the same difference, negative for every n
Constant sequence
$a_{n+1} - a_n = 0$
every term equal

The sequence a(n) = 2n + 1 for n from 1 to 7. The graph of a sequence is a set of separate dots — there is nothing between them, because there is no natural number between 3 and 4.

The sequence b(n) = 10 − 2n. The dots step down by the same amount every time, so the sequence is decreasing.

Frequently asked questions

A luminous parabola and a straight line crossing above a faint coordinate grid, the vertex flaring like a star, against a cosmic nebula in violet and magenta

Functions and analysis

Why a function's graph can never double back

The whole definition of a function comes down to one word: uniqueness. One argument, one value — and not a single one more. That seemingly minor condition decides why a circle is not the graph of a function, why a line and a parabola describe so much of the world, and why four datasets with identical summary statistics can look nothing alike.

12 min