Sequences

In Grade 6 you met two ways to describe a sequence. A term-to-term rule tells you how to step from one term to the next — for $3, 7, 11, 15, \ldots$ the rule is "add 4". A position-to-term rule works out any term straight from its position number $n$.

This year you go deeper. You will find the nth-term rule of any linear sequence quickly, use it to answer "which term is this?" questions, and handle sequences that go down as well as up.

The big idea this year is fluency. You should be able to look at $7, 11, 15, 19, \ldots$ and write its rule, $4n + 3$, almost on sight. That speed frees your brain for the more interesting questions — like whether 403 is a term of that sequence.

Keep your Grade 6 toolkit handy; this guide builds straight on top of it.

A linear sequence climbs or falls by the same amount each time — the common difference $d$. Its nth-term rule always has the shape $dn + c$.

Here is the reliable method for $5, 11, 17, 23, \ldots$:

1. Find the common difference: $11 - 5 = 6$, so the rule starts with $6n$.
2. List what $6n$ gives: $6, 12, 18, 24, \ldots$
3. Compare with the real sequence: each term is 1 less than $6n$.
4. So the rule is $6n - 1$.

A faster shortcut: the value of $c$ is the term you would get before the first term — the "zeroth" term. For $5, 11, 17, \ldots$ step back from 5 by 6 to get $-1$, which is exactly the $c$ in $6n - 1$.

Always check by substituting $n = 1$: $6 \times 1 - 1 = 5$. Correct first term, so the rule is sound.

Not every sequence climbs. The sequence $20, 17, 14, 11, \ldots$ falls by 3 each time, so its common difference is $-3$. The nth-term rule starts with $-3n$.

The method is exactly the same — you just carry the negative sign carefully:

1. Common difference is $-3$, so the rule starts with $-3n$.
2. List what $-3n$ gives: $-3, -6, -9, -12, \ldots$
3. Compare: each real term is 23 more than $-3n$ (since $20 - (-3) = 23$).
4. So the rule is $-3n + 23$.

Check with $n = 1$: $-3 \times 1 + 23 = 20$. The first term is right.

A decreasing sequence will eventually pass through zero and into negative numbers. Using the rule, you can find exactly where. For $-3n + 23$, the term equals zero when $-3n + 23 = 0$, which is not a whole number — so this sequence skips straight over zero.

Here is where the nth-term rule really earns its place. Instead of asking "what is term 20?", you can ask "which term is 87?" — or "is 87 even in the sequence at all?".

The trick is to work backwards. Set the nth-term rule equal to the number, then solve the equation for $n$.

Take the sequence $4, 7, 10, 13, \ldots$ with rule $3n + 1$. Is 87 a term?

$3n + 1 = 87$ $3n = 86$ $n = 86 \div 3 = 28.67\ldots$

Because $n$ is not a whole number, 87 is not a term of this sequence.

Now test 88: $3n + 1 = 88$ gives $n = 29$. That is a whole number, so 88 is the 29th term.

The rule: a number belongs to a linear sequence only when solving its equation gives a positive whole counting number for $n$.

Not all sequences are linear. Several famous ones have differences that keep changing, so no $dn + c$ rule can describe them.

• Square numbers: $1, 4, 9, 16, 25, \ldots$ — rule $n^2$. The first differences are the odd numbers.
• Triangular numbers: $1, 3, 6, 10, 15, \ldots$ — each gap is one larger than the last.
• Cube numbers: $1, 8, 27, 64, \ldots$ — rule $n^3$.
• Fibonacci: $1, 1, 2, 3, 5, 8, 13, \ldots$ — each term is the sum of the two before it.

A powerful Stage 8 tool is the second difference — the difference of the differences. For a linear sequence the first differences are already constant. For square or triangular numbers the first differences change but the second difference is constant, which is the fingerprint of a sequence that grows in a quadratic way.

So before reaching for $dn + c$, check the differences. Constant first difference means linear; constant second difference means something more interesting.

The nth-term skills you sharpen now open doors across maths:

• Algebra treats the nth-term rule as an expression — substituting and solving are the same skills you use everywhere else.
• Linear graphs plot a linear sequence as evenly spaced points on a straight line; the common difference becomes the gradient.
• Functions are close cousins — a position-to-term rule is a function with the position as its input.
• In everyday life, sequences model savings that grow by a fixed amount, loan repayments and patterns in nature and design.

Later you will meet quadratic sequences, where the second difference you have just met becomes the key to the rule. The groundwork is being laid right here.

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