Sequences

A sequence is a list of numbers in a set order, following some pattern. Each number in the list is called a term.

You meet sequences all the time: the dates on a calendar, the page numbers in a book, the seats numbered along a row.

The sequence above is $3, 7, 11, 15, 19, \ldots$ The three dots mean "and it carries on".

The order matters. The 1st term is 3, the 2nd term is 7, and so on. The number that says where a term sits is its position.

The pattern is the heart of a sequence. Once you spot the pattern, you can continue the list, fill in gaps, and even leap straight to a term far down the line. The rest of this guide is about describing that pattern in two clever ways.

The simplest way to describe a sequence is a term-to-term rule — it tells you what to do to one term to get the next.

For $3, 7, 11, 15, \ldots$ the rule is "add 4". Each term is 4 more than the one before.

Term-to-term rules can use any operation:

• $2, 4, 6, 8, \ldots$ — rule: add 2.
• $20, 17, 14, 11, \ldots$ — rule: subtract 3.
• $2, 4, 8, 16, \ldots$ — rule: multiply by 2.
• $1, 3, 9, 27, \ldots$ — rule: multiply by 3.

To continue a sequence with a term-to-term rule, you just keep applying it. To take $3, 7, 11, 15$ further, add 4: the next terms are 19, 23, 27.

A term-to-term rule always needs a starting term. "Add 4" on its own does not pin down a sequence — you must also know it begins at 3.

There is one catch. A term-to-term rule is great for the next term, but slow for a far-off one. To find the 50th term of $3, 7, 11, \ldots$ you would have to add 4 forty-nine times. That is exactly the problem the next idea solves.

A position-to-term rule is much faster. It works out any term directly from its position number — no need to list every term before it.

We use the letter $n$ for the position: the 1st term is $n = 1$, the 2nd is $n = 2$, and so on.

Take the rule $\text{term} = 3n + 2$. To find a term, substitute its position for $n$:

• 1st term: $3 \times 1 + 2 = 5$.
• 2nd term: $3 \times 2 + 2 = 8$.
• 3rd term: $3 \times 3 + 2 = 11$.

This is the power of a position-to-term rule. Want the 50th term? Just put $n = 50$: $3 \times 50 + 2 = 152$. One quick calculation, no listing.

A position-to-term rule is really a function — the position goes in, the term comes out.

A linear sequence goes up (or down) by the same amount each time. That steady step is the common difference.

The position-to-term rule for a linear sequence is called the nth-term rule, and there is a neat method to find it.

For the sequence $5, 8, 11, 14, \ldots$:

1. Find the common difference: it goes up by 3 each time, so the rule starts with $3n$.
2. List what $3n$ gives: $3, 6, 9, 12, \ldots$
3. Compare with the real sequence: each term is 2 more than $3n$.
4. So the nth-term rule is $3n + 2$.

The shortcut: the common difference is the number in front of $n$, then you add or subtract a number to match the first term.

Check it by putting $n = 1$: $3 \times 1 + 2 = 5$ — the correct first term.

A few sequences turn up so often that they are worth knowing on sight.

• Square numbers: $1, 4, 9, 16, 25, \ldots$ — found by squaring the position, so the rule is $n^2$.
• Triangular numbers: $1, 3, 6, 10, 15, \ldots$ — each adds one more than the last ($+2$, then $+3$, then $+4$).
• Cube numbers: $1, 8, 27, 64, \ldots$ — found by cubing the position, rule $n^3$.
• The Fibonacci sequence: $1, 1, 2, 3, 5, 8, 13, \ldots$ — each term is the sum of the two before it. It even appears in nature, in flower petals and pine cones.

Square, triangular and cube numbers are not linear — their differences are not constant, so they do not have a simple $dn + c$ rule. Recognising them quickly will save you a lot of time.

Getting confident with sequences now helps across many topics:

• Algebra uses the nth-term rule — it is just an expression, so simplifying and substituting skills apply.
• Graphs can plot a sequence as points; a linear sequence forms a straight line.
• Functions are closely linked — a position-to-term rule is a function with the position as its input.
• In everyday life, sequences describe savings that grow by a fixed amount, repeating timetables and patterns in nature.

Pattern-spotting is a skill in itself, and it sharpens your number sense for problem-solving everywhere.

If sequence work feels hard later, it is usually finding the common difference that needs a refresh. Come back to this guide whenever you need it — there is no prize for rushing.