HookA 13th-century rabbit problem is hiding in every sunflower
In 1202, an Italian merchant's son called Leonardo of Pisa — later nicknamed Fibonacci — published a book called Liber Abaci, and buried in it was a toy puzzle: if a pair of rabbits produces a new pair every month, and each new pair starts breeding after one month, how many pairs are there each month? Counting them out gives 1, 1, 2, 3, 5, 8, 13, 21, 34 — each number the sum of the two before it. Fibonacci meant it as an exercise. He could not have known that the same run of numbers governs the spirals of seeds in a sunflower head, the bracts on a pinecone, and the branching of countless plants — because packing seeds in Fibonacci spirals is the most efficient way nature has found to fill a circle without gaps.
That is the quiet power of a sequence: a short rule generates an endless, predictable list, and a list you can predict is a list you can control. This section is about spotting the rule behind a run of numbers and using it. You will describe sequences two different ways — step by step, and by jumping straight to any position — recognise the famous families (square, triangular, cube, arithmetic, geometric and Fibonacci-type numbers), and build the algebraic 'nth term' formula that lets you find the 100th term without writing out the first 99. It is examined on both the calculator and non-calculator papers, and the nth-term skill is one of the most reliable sources of method marks in the whole Foundation course.
ModelTwo ways to describe a sequence
There are exactly two ways to pin down a sequence, and telling them apart is half the battle. A term-to-term rule says how to get from one term to the next: 'start at 4 and add 3 each time' gives 4, 7, 10, 13, 16. It is easy to use but has a weakness — to reach the 50th term you must grind through all 49 before it, because each term depends on the one in front.
A position-to-term rule fixes that by giving a formula in terms of the position number n: term = 3n + 1 lets you drop any position straight in. Want the 50th term? Substitute n = 50: 3 × 50 + 1 = 151, no grinding required. The position n is always a whole counting number — 1 for the first term, 2 for the second — so you never substitute n = 0 or a fraction. This position rule is what the exam calls the 'nth term', and it is the single most useful idea in the section.
The practical skill is generating terms cleanly from whichever rule you are handed. Given a term-to-term rule, apply it repeatedly and write each result. Given a position-to-term rule, substitute n = 1, 2, 3, 4 in turn. The classic Foundation slip is muddling the two — using the previous term when the rule wants the position number, or vice versa — so read carefully which kind of rule the question has given you before you start.
A sequence has first term 4 and the term-to-term rule 'multiply by 2, then subtract 1'. Write the first five terms. Start at 4. Next: 4 × 2 − 1 = 7. Next: 7 × 2 − 1 = 13. Next: 13 × 2 − 1 = 25. Next: 25 × 2 − 1 = 49. So the sequence is 4, 7, 13, 25, 49. Notice you could not have jumped straight to the fifth term — each one needed the one before it, which is exactly the limitation a position-to-term rule removes. Compare a position rule: the sequence given by 2n + 3 has terms 5, 7, 9, 11 (substituting n = 1, 2, 3, 4), and its 100th term is instant: 2 × 100 + 3 = 203.
MechanismThe famous families — know them on sight
AQA expects instant recognition of several special sequences, because questions often hinge on naming them. Square numbers come from multiplying a whole number by itself: 1, 4, 9, 16, 25, 36 (that is 1², 2², 3²…). Cube numbers multiply three times: 1, 8, 27, 64, 125 (1³, 2³, 3³…). Triangular numbers count dots stacked in a growing triangle — 1, 3, 6, 10, 15 — where you add 2, then 3, then 4, then 5; each new row adds one more dot than the last.
An arithmetic sequence (also called a linear sequence) goes up or down by a fixed step called the common difference: 5, 8, 11, 14 adds 3 each time; 20, 17, 14, 11 subtracts 3. A geometric sequence instead multiplies by a fixed common ratio: 3, 6, 12, 24 doubles; 80, 40, 20, 10 halves. And a Fibonacci-type sequence adds the two previous terms, like the rabbits: 1, 1, 2, 3, 5, 8, or any starting pair such as 4, 7, 11, 18, 29.
The skill that ties them together is identification: to name an unknown sequence, test it. A constant difference between terms means arithmetic. A constant ratio (each term divided by the one before gives the same number) means geometric. Neither, but a constant second difference, means quadratic — like the square numbers. And each term being the sum of the previous two means Fibonacci-type. Running that quick diagnostic tells you what you are looking at before you try to continue it.
Identify and continue 2, 5, 10, 17, 26. First differences: 5 − 2 = 3, 10 − 5 = 5, 17 − 10 = 7, 26 − 17 = 9 — these are not constant, so it is not arithmetic. Check the ratios: 5 ÷ 2 = 2.5 but 10 ÷ 5 = 2 — not constant, so not geometric. Now the second differences: 5 − 3 = 2, 7 − 5 = 2, 9 − 7 = 2 — constant, so this is a quadratic sequence. To continue it, the next first difference is 9 + 2 = 11, so the next term is 26 + 11 = 37. (In fact this sequence is n² + 1, but at Foundation you continue a quadratic by its second differences rather than finding a formula for it.)
MechanismSequences from patterns and diagrams
A favourite Foundation question hands you a growing pattern of matchsticks, tiles or dots and asks for the number in the 10th, 50th or nth arrangement. The reliable method is to count the first few patterns, tabulate the results as a sequence, find the rule, and only then leap ahead. Trying to draw the 50th pattern is a trap; finding its rule is the point.
Take a row of squares made from matchsticks: one square uses 4 matches, two squares in a row use 7, three use 10. Tabulated, that is the sequence 4, 7, 10, 13 — arithmetic, common difference 3, because every extra square adds 3 matches (it shares one side with its neighbour). Once you have the sequence, the pattern problem becomes an ordinary nth-term problem, which the next block solves.
Why the difference equals 3 is worth understanding, not just spotting: the first square needs 4 sides, but each square after it only needs 3 new matches because it borrows the fourth from the square before. That structural reason is the bridge to the formula — the constant '3 more each time' becomes the '3n' in the nth-term rule, and the leftover matches become the adjustment. Reading the growth of a picture and turning it into a numerical sequence is the exact skill the question rewards.
A pattern grows: pattern 1 has 5 tiles, pattern 2 has 8 tiles, pattern 3 has 11 tiles. How many tiles in pattern 20? Tabulate the sequence: 5, 8, 11, 14… with a common difference of 3. This is arithmetic, so it has an nth-term rule of the form 3n + something. Since 3 × 1 = 3 but the first term is 5, the adjustment is +2, giving 3n + 2. Now jump to pattern 20 without drawing it: 3 × 20 + 2 = 62 tiles. Check the rule on a known term first — 3 × 3 + 2 = 11, which matches pattern 3 — so the formula is trustworthy before you rely on it for pattern 20.
MechanismThe nth term of a linear sequence — the workhorse formula
Finding the nth-term rule of an arithmetic (linear) sequence is the section's headline skill, and it has a fixed two-step method. Step one: find the common difference and put it in front of n — that is the coefficient of n. Step two: work out the adjustment by comparing your dn against the actual first term. For 5, 8, 11, 14, the common difference is 3, so start with 3n. But 3n gives 3, 6, 9, 12 (the three-times table), while the real sequence is 5, 8, 11, 14 — each term is 2 bigger. So the rule is 3n + 2.
Why the common difference becomes the coefficient of n: a linear sequence adds the same amount every step, and 3n is precisely 'the 3-times table', which also adds 3 every step. Your sequence marches in lockstep with the 3-times table, just shifted up or down by a constant — and that constant shift is the number you add or subtract. A downhill sequence works identically with a negative coefficient: 20, 17, 14, 11 has common difference −3, so it starts −3n, and since −3n gives −3, −6, −9 while the sequence is 20, 17, 14, the adjustment is +23, giving −3n + 23.
The formula then answers 'is this number in the sequence?' questions, a favourite of examiners. Set the rule equal to the target and solve for n; if n comes out a whole number, the term is in the sequence, and if it does not, it is not. (Finding the nth term of a quadratic sequence is Higher tier only — at Foundation you continue quadratics by their second differences but are not asked for their formula.)
Find the nth term of 5, 8, 11, 14, then decide whether 100 is a term. Common difference is 3, so begin with 3n. The 3-times table 3, 6, 9, 12 is 2 short of 5, 8, 11, 14 each time, so add 2: the nth term is 3n + 2. Check: n = 1 gives 3 + 2 = 5 ✓, n = 4 gives 12 + 2 = 14 ✓. Now test 100 by solving 3n + 2 = 100: subtract 2 for 3n = 98, divide by 3 for n = 32.67. That is not a whole number, so 100 is not in the sequence. By contrast 98 gives 3n + 2 = 98, so 3n = 96, n = 32 — a whole number, so 98 is the 32nd term. Showing that n is not whole is exactly the reasoning the mark scheme wants for 'explain why'.
DataGeometric growth — when multiplying runs away with you
A geometric sequence multiplies by a fixed common ratio each step, and because repeated multiplying compounds, it grows or shrinks far faster than any arithmetic sequence. The old legend of the inventor who asked a king for one grain of rice on the first square of a chessboard, two on the second, four on the third, doubling all the way, is a geometric sequence with ratio 2 — and by the 64th square it demands more rice than has ever been grown on Earth, because doubling 63 times reaches astronomically large numbers.
To work with one, find the ratio by dividing any term by the one before it: for 3, 6, 12, 24 the ratio is 6 ÷ 3 = 2, confirmed by 12 ÷ 6 = 2. Continue by multiplying: the next term is 24 × 2 = 48. Foundation restricts these to a rational ratio greater than zero, so the ratio might be 2, or 3, or a fraction like ½ (a halving sequence such as 80, 40, 20, 10), or even 1.5 — but never a surd, which is Higher tier. A halving sequence is still geometric; the ratio is simply less than 1, so the terms shrink.
The Fibonacci sequence from the introduction is not geometric — it adds rather than multiplies — but it has a beautiful link: divide each Fibonacci term by the one before and the answers home in on roughly 1.618, the 'golden ratio' that artists and architects have chased for centuries. You are not asked to prove that at Foundation, but recognising that Fibonacci-type sequences add the previous two terms, while geometric sequences multiply by a fixed ratio, is exactly the distinction the exam tests.
A geometric sequence starts 2, 6, 18, 54. Find the common ratio and the next two terms. Ratio = 6 ÷ 2 = 3, checked by 18 ÷ 6 = 3 and 54 ÷ 18 = 3 — a constant ratio of 3, so it is geometric. Continue by multiplying: 54 × 3 = 162, then 162 × 3 = 486. So the sequence continues 162, 486. Contrast a Fibonacci-type sequence starting 4, 7: add the previous two each time — 4, 7, 11 (4 + 7), 18 (7 + 11), 29 (11 + 18), 47. One multiplies, the other adds; naming which is which is the mark.
VocabularyKey terms the mark scheme pays for
TrapsMisconceptions that cost marks
ExamWhat examiners want
Sequences appear on both papers, and the nth-term question is one of the most predictable mark-earners in the Foundation course, so drill the two-step method until it is automatic: write the common difference as the coefficient of n, then compare against the actual terms to fix the adjustment — and show both steps, because the 'dn' line is a method mark even if the constant is wrong. Always verify your rule on a known term (substitute n = 1 and n = 2) before you trust it for a distant term; a formula that fails the first term is wrong, and the check costs seconds.
Read the command carefully. 'Write down the next two terms' wants a term-to-term step; 'find the nth term' or 'find an expression for the nth term' wants the position formula in n; and 'is 100 a term in this sequence?' wants you to set the rule equal to 100 and solve — if n is not a whole number, say so explicitly, because that sentence is the mark. For pattern questions, tabulate the first three arrangements into a sequence before generalising, rather than trying to draw large cases.
Know the named families cold, because naming them is often worth a mark on its own: square, cube and triangular numbers, arithmetic versus geometric (constant difference versus constant ratio), and Fibonacci-type (add the previous two). If a sequence has no constant first difference or ratio, check the second difference — a constant one signals a quadratic sequence, which at Foundation you continue by extending those second differences, never by attempting an nth-term formula (that is Higher tier). Keep negatives intact when a sequence descends: a common difference of −3 makes the rule start −3n, and dropping the sign is the commonest error on decreasing sequences.