What Is a Sequence?
A sequence is an ordered list of numbers (or other objects) that usually follows a rule, where each number in the list is called a term. The sequence $3, 6, 9, 12, \ldots$ is ordered, and its rule — add $3$ each step — tells you how to get from one term to the next.
Two features make a sequence a sequence:
Order matters. Unlike a set, where ${1, 2}$ and ${2, 1}$ are the same, the sequences $1, 2$ and $2, 1$ are different. Position is part of the identity.
There is usually a rule. Most sequences follow a pattern that lets you find any term — the next one, or the hundredth. Each entry is a term, and the rule connects them.
A sequence can be finite (a fixed number of terms, like $2, 4, 6, 8$) or infinite (continuing without end, written with an ellipsis: $2, 4, 6, 8, \ldots$).
What Are the Types of Sequences?
What kinds of sequences are there? Several, and each is defined by the rule that links its terms. The most common ones every student meets:
Arithmetic sequence — each term is found by adding a fixed number (the common difference). $2, 5, 8, 11$ adds $3$ each step.
Geometric sequence — each term is found by multiplying by a fixed number (the common ratio). $3, 6, 12, 24$ multiplies by $2$ each step.
Fibonacci sequence — each term is the sum of the two before it: $0, 1, 1, 2, 3, 5, 8, 13, \ldots$
Harmonic sequence — the reciprocals form an arithmetic sequence: $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots$
Triangular, square, and cube number sequences — built from $n(n+1)/2$, $n^2$, and $n^3$: the squares run $1, 4, 9, 16, 25, \ldots$
That is the working set students meet first; a few others (such as quadratic sequences) appear later. For the arithmetic type in full — every formula and a worked deep-dive, and for the broader algebra treatment of all types, see sequences in algebra.
Type | Rule | Example |
|---|---|---|
Arithmetic | Add a fixed number | $2, 5, 8, 11, \ldots$ |
Geometric | Multiply by a fixed number | $3, 6, 12, 24, \ldots$ |
Fibonacci | Add the two previous terms | $0, 1, 1, 2, 3, 5, \ldots$ |
Square numbers | $n^2$ | $1, 4, 9, 16, \ldots$ |
Examples of Sequences
The set runs from naming a rule to finding a distant term with a formula and reading the order of a real-world sequence. Read each problem first.
Example 1
What is the rule for the sequence $5, 10, 15, 20, \ldots$, and what is the next term?
Each term is $5$ more than the last, so the rule is "add $5$." The next term after $20$ is:
$$20 + 5 = 25$$
This is an arithmetic sequence with common difference $5$.
Example 2
This is the slip that costs the most marks on the nth-term formula, so it is worth the wrong road first.
Find the 10th term of the arithmetic sequence $3, 7, 11, 15, \ldots$
The tempting wrong move. A student knows the first term is $3$ and the gap is $4$, and writes "10th term $= 3 + 10 \times 4 = 43$." It feels right — first term plus ten gaps.
Why it breaks. Getting from the 1st term to the 10th means taking only nine steps, not ten — you do not add a gap to reach the term you already start on. Adding ten gaps overshoots by one step.
The correct method. The nth-term rule is $a_n = a_1 + (n-1)d$, with first term $a_1 = 3$, common difference $d = 4$, and $n = 10$:
$$a_{10} = 3 + (10 - 1) \times 4 = 3 + 36 = 39$$
The 10th term is 39. The $(n-1)$ is doing the real work — it counts the steps between terms, not the terms themselves.
Example 3
Find the rule for the geometric sequence $2, 6, 18, 54, \ldots$ and the next term.
Each term is the previous one times $3$ (the common ratio), so the rule is "multiply by $3$." The next term after $54$ is:
$$54 \times 3 = 162$$
In a geometric sequence the terms grow by multiplication, which is why they climb so much faster than an arithmetic one.
Example 4
The square-number sequence is $1, 4, 9, 16, \ldots$ Use the rule $a_n = n^2$ to find the 7th term.
The rule gives any term directly from its position. For $n = 7$:
$$a_7 = 7^2 = 49$$
The 7th term is 49. A rule like $a_n = n^2$ is powerful because it jumps straight to any term without listing all the ones before it.
Example 5
Is $0, 1, 1, 2, 3, 5, 8, \ldots$ arithmetic, geometric, or something else? Find the next term.
The differences ($1, 0, 1, 1, 2, 3$) are not constant, so it is not arithmetic; the ratios are not constant either, so it is not geometric. Each term is the sum of the two before it — this is the Fibonacci sequence. The next term is:
$$5 + 8 = 13$$
Example 6
A theatre has $20$ seats in the front row, $22$ in the next, $24$ in the next, and so on. How many seats are in the 8th row?
The seat counts form an arithmetic sequence: first term $a_1 = 20$, common difference $d = 2$. Using $a_n = a_1 + (n-1)d$ with $n = 8$:
$$a_8 = 20 + (8 - 1) \times 2 = 20 + 14 = 34$$
The 8th row has 34 seats. The same nth-term rule that handles abstract sequences answers a real seating-plan question — and notice it relies on the $(n-1)$ step count, exactly the point Example 2 rescued.
Why Sequences Are Worth Learning
"One introduction, and the rabbits never stopped."
In 1859, a settler released $24$ rabbits into the Australian wild for sport hunting. With no natural predators, the population followed a roughly geometric sequence — multiplying each season — until hundreds of millions of rabbits stripped farmland across the continent. The Australian rabbit plague is what unchecked geometric growth looks like outside a textbook: a sequence that doubles and doubles again outruns intuition fast.
That is the reason sequences matter — they are how mathematics describes anything that grows or repeats by a rule:
Growth and decay. Populations, compound interest, and radioactive decay all follow geometric sequences. The difference between adding and multiplying each step is the difference between a savings account and a runaway debt.
Patterns in nature. The Fibonacci sequence shows up in the spirals of sunflowers, pinecones, and shells — a rule simple enough to write in one line, visible across the living world.
Computer science. Sequences underlie how programs loop, how data is indexed, and how algorithms step through tasks in order.
Calculus, later on. Whether an infinite sequence settles toward a single value — converges — is the gateway idea to limits and series, the foundation of calculus.
The deeper point: a sequence turns "what comes next?" from a guess into a calculation. Once you have the rule, you are no longer continuing a pattern by feel — you are reading the structure that was there all along.
Where Students Trip Up on Sequences
Mistake 1: Adding the wrong number of steps in the nth-term formula
Where it slips in: Using $a_n = a_1 + n \times d$ instead of $a_n = a_1 + (n-1)d$.
Don't do this: Add $n$ common differences to the first term — it overshoots by one step.
The correct way: Use $(n-1)$, because reaching the nth term takes $(n-1)$ steps from the first. For the 10th term: nine steps, not ten.
Mistake 2: Confusing arithmetic with geometric
Where it slips in: Seeing a growing sequence and assuming it is arithmetic without checking.
Don't do this: Treat $3, 6, 12, 24$ as "add the same number" when the gap keeps changing.
The correct way: Check the relationship. Constant difference between terms means arithmetic; constant ratio means geometric. $3, 6, 12, 24$ has a constant ratio of $2$, so it is geometric.
Mistake 3: Treating a sequence like an unordered set
Where it slips in: Assuming $1, 2, 3$ and $3, 2, 1$ are "the same numbers, so the same sequence."
Don't do this: Ignore order when comparing or listing sequences.
The correct way: In a sequence, position is part of the identity. $1, 2, 3$ and $3, 2, 1$ are different sequences, even though they use the same numbers.
The Short Version
A sequence is an ordered list of numbers that follows a rule, with each number called a term.
Order is part of the identity — $1, 2, 3$ and $3, 2, 1$ are different sequences.
The main types are arithmetic (add), geometric (multiply), and Fibonacci (sum the two before), among others.
The nth-term rule $a_n = a_1 + (n-1)d$ finds any term — the $(n-1)$ counts steps, not terms.
Sequences describe anything that grows or repeats by a rule, from compound interest to the spirals of a sunflower.
Practice These Before Moving On
Work through these three, then check against the article.
What type of sequence is $5, 10, 20, 40, \ldots$, and what is the next term?
Find the 12th term of the arithmetic sequence $4, 9, 14, 19, \ldots$
Write the first five terms of the sequence given by the rule $a_n = 3n - 1$.
If problem 2 caught you out, reread Example 2 — reaching the 12th term takes eleven steps, so use $(n-1)$. Want a live Bhanzu trainer to walk your child through sequences? Book a free demo class — online globally.
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