Sequences

In Grade 7 you became fluent with linear sequences. You can take $7, 11, 15, 19, \ldots$, spot the common difference of 4, and write the nth-term rule $4n + 3$ almost on sight.

This year you go further. You will meet quadratic sequences — sequences whose rule contains an $n^2$ term — and learn the elegant method that unlocks them.

The big idea this year is analysis. Before you write any rule, you study the differences between terms. A constant first difference means linear. A first difference that itself changes steadily points to something new and more interesting.

Keep your Grade 7 toolkit close — every linear skill still applies, and quadratics build straight on top of it.

To generate a sequence, you take its nth-term rule and substitute the positions $n = 1, 2, 3, \ldots$ one after another.

For the linear rule $5n - 2$:

• $n = 1$: $5 \times 1 - 2 = 3$
• $n = 2$: $5 \times 2 - 2 = 8$
• $n = 3$: $5 \times 3 - 2 = 13$

So the sequence is $3, 8, 13, 18, \ldots$

Generating a quadratic sequence works the same way — you just take care with the squaring. For the rule $n^2 + 1$:

Be careful with the order of operations: in $n^2 + 1$ you square first, then add. For $n = 3$ that is $9 + 1 = 10$, not $(3 + 1)^2 = 16$.

Generating a sequence from its rule is also a brilliant way to check a rule you have just found.

The most powerful tool for analysing a sequence is its differences.

The first difference is the gap between neighbouring terms. The second difference is the difference of those first differences.

Here is the rule of thumb that drives all your analysis:

• If the first difference is constant, the sequence is linear ($dn + c$).
• If the second difference is constant, the sequence is quadratic (it has an $n^2$ term).
• If neither is constant, the sequence is something else again.

For $2, 5, 10, 17, 26, \ldots$ the first differences are $3, 5, 7, 9$ — not constant. But the second differences are $2, 2, 2$ — constant. So this is a quadratic sequence.

Always check the differences before you decide what kind of rule to look for. It saves you chasing a linear rule that can never fit.

Once a sequence has a constant second difference, there is a neat method to find its rule.

Step one — the number in front of $n^2$ is half the second difference. For a second difference of 2, the rule starts with $1n^2$, that is $n^2$.

Step two — work out what $n^2$ gives, then subtract it from the sequence to see what is left.

Take the quadratic sequence $4, 7, 12, 19, 28, \ldots$:

1. The second differences are $2$, so the rule starts with $n^2$.
2. List what $n^2$ gives: $1, 4, 9, 16, 25, \ldots$
3. Subtract: each term is $3$ more than $n^2$.
4. So the rule is $n^2 + 3$.

Check it with $n = 1$: $1^2 + 3 = 4$ — the correct first term. The leftover part is often a constant, but sometimes it is itself a small linear sequence in $n$.

Putting it all together, here is how to analyse any sequence you meet.

1. Write the first differences. Constant? It is linear — find $dn + c$.
2. If not, write the second differences. Constant? It is quadratic — halve it for the $n^2$ term.
3. If neither is constant, check for a familiar special sequence such as cubes or Fibonacci.

Once you have the rule, it becomes a problem-solving tool. The rule lets you jump straight to far-off terms and answer "which term?" questions by working backwards.

For example, with the quadratic rule $n^2 + 3$, the 10th term is simply $10^2 + 3 = 103$ — no listing of 10 terms required.

Analysing first, then using the rule, is the heart of confident Stage 9 sequence work.

The sequence analysis skills you sharpen now feed straight into the maths ahead:

• Algebra treats every nth-term rule as an expression — including quadratic ones with $n^2$ terms.
• Quadratic graphs plot a quadratic sequence as a smooth curve rather than a straight line.
• Functions are close cousins — a position-to-term rule is a function with the position as its input.
• In everyday life, quadratic sequences describe areas of growing squares, the path of a thrown ball, and patterns that speed up.

Later you will study quadratic expressions and equations in depth, and the $n^2$ rules you find here will feel completely familiar.

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