Sequences

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.

• A linear sequence has a constant first difference.
• A quadratic sequence has a rule with an $n^2$ term.
• Stage 9 focuses on analysing differences before choosing a rule.
• Quadratic work builds directly on your linear sequence skills.

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.

• Substitute $n = 1, 2, 3, \ldots$ to generate any sequence.
• Quadratic rules are generated the same way as linear ones.
• Square the position before adding or subtracting.
• Generating terms is a quick way to check a rule.

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.

• The first difference is the gap between neighbouring terms.
• The second difference is the difference of the first differences.
• Constant first difference means a linear sequence.
• Constant second difference means a quadratic sequence.

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 $n$.

• The $n^2$ coefficient is half the second difference.
• List the $n^2$ values, then subtract from the sequence.
• What is left over completes the rule.
• Always check the finished rule on the first term.

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.

• Check first differences, then second differences, in order.
• A constant first difference means linear; constant second means quadratic.
• Use the rule to jump to far-off terms instantly.
• Work backwards with the rule to answer 'which term' questions.

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.

• Quadratic nth-term rules are expressions with $n^2$.
• A quadratic sequence plots as a curve, not a straight line.
• A position-to-term rule is a kind of function.
• Quadratic sequences model areas, growth and motion.

Square the position first, then add 2. For $n = 1$: