Sequences: Linear and Quadratic

A sequence is an ordered list of numbers, called terms, that follow a rule. This lesson covers describing a sequence by a term-to-term rule and by a position-to-term rule, finding the $n$nth term of a linear (arithmetic) sequence, and — at Higher tier — finding the $n$nth term of a quadratic sequence. It also covers the special sequences you are expected to recognise, including Fibonacci-type and geometric sequences. Generating terms and finding rules is AO1; spotting structure and justifying a rule is AO2, and applying sequences to a context is AO3.

Key Vocabulary

Term	Meaning
Sequence	An ordered list of numbers following a rule
Term	A single number in a sequence
Term-to-term rule	How to get from one term to the next
Position-to-term rule	A formula giving the term in terms of its position $n$n
$n$nth term	The formula for the term in position $n$n
Common difference	The constant gap $d$d between terms of a linear sequence
Common ratio	The constant multiplier $r$r between terms of a geometric sequence
Arithmetic / linear sequence	A sequence with a constant common difference

Sequences appear throughout mathematics and the world around us — the rows of seats in a theatre, the interest added to savings each year, the patterns in tilings and stacks. The skill the examiner is testing is twofold: spotting the rule that generates a list, and then expressing that rule precisely enough to predict any term you like. The most powerful tool is the $n$nth term formula, because it lets you find, say, the $100$100th term without writing out the first ninety-nine.

Term-to-Term and Position-to-Term Rules

A term-to-term rule describes how each term is obtained from the previous one — for example "start at $4$4 and add $3$3 each time". A position-to-term rule (the $n$nth term) lets you jump straight to any term using its position number $n$n — for example "the $n$nth term is $3n + 1$3n+1". The crucial difference is that a term-to-term rule needs the previous term to work, whereas a position-to-term rule works directly from the position number. OCR questions often ask for one or the other specifically, so read the command carefully.

Worked Example 1

A sequence starts $4, 7, 10, 13, \dots$4,7,10,13,… Describe the term-to-term rule and write the next two terms.

Each term is $3$3 more than the one before, so the rule is "add $3$3". The next two terms are $16$16 and $19$19.

Answer: add $3$3; next terms $16$16 and $19$19.

Worked Example 2

The $n$nth term of a sequence is $5n - 2$5n−2. Work out the first three terms.

Substitute $n = 1, 2, 3$n=1,2,3: $5(1) - 2 = 3$5(1)−2=3, $5(2) - 2 = 8$5(2)−2=8, $5(3) - 2 = 13$5(3)−2=13.

Answer: $3, 8, 13$3,8,13

Common error: reading $5n - 2$5n−2 as the term-to-term rule. It is a position-to-term rule — substitute the position number, do not just add $5$5.

The $n$nth Term of a Linear Sequence

A linear (arithmetic) sequence has a constant common difference $d$d. Its $n$nth term has the form $dn + (\text{first term} - d)$dn+(first term−d) — in practice, write $dn$dn first, then adjust by a constant so the formula matches term $1$1.

Worked Example 3

Find the $n$nth term of $7, 11, 15, 19, \dots$7,11,15,19,…

The common difference is $4$4, so the $n$nth term starts $4n$4n. Checking $n = 1$n=1: $4(1) = 4$4(1)=4, but the first term is $7$7, so add $3$3: $n$nth term $= 4n + 3$=4n+3.

Answer: $4n + 3$4n+3

Worked Example 4

Find the $n$nth term of $20, 17, 14, 11, \dots$20,17,14,11,…

The common difference is $-3$−3, so begin $-3n$−3n. At $n = 1$n=1: $-3(1) = -3$−3(1)=−3, but the first term is $20$20, so add $23$23: $n$nth term $= -3n + 23$=−3n+23, often written $23 - 3n$23−3n.

Answer: $23 - 3n$23−3n

Worked Example 5 — using the $n$nth term

The $n$nth term of a sequence is $6n - 1$6n−1. Is $200$200 a term in this sequence?

Set $6n - 1 = 200$6n−1=200: $6n = 201$6n=201, so $n = 33.5$n=33.5. Because $n$n must be a positive integer, $200$200 is not a term.

Answer: no, because $n = 33.5$n=33.5 is not a whole number.

Common error: concluding "$200$200 is a term" without checking that $n$n comes out as a whole number.

Worked Example 5b

The $n$nth term of a sequence is $50 - 4n$50−4n. Work out the first term that is negative.

We need $50 - 4n < 0$50−4n<0, i.e. $50 < 4n$50<4n, so $n > 12.5$n>12.5. The first whole-number position satisfying this is $n = 13$n=13, giving $50 - 4(13) = 50 - 52 = -2$50−4(13)=50−52=−2.

Answer: the $13$13th term, which is $-2$−2.

This blends sequences with inequalities — a common combination at the upper end of the Foundation tier and the lower end of Higher. Setting up an inequality from the $n$nth term, then taking the first integer position, is the key technique.

Quadratic Sequences [Higher]

This whole section is Higher tier. A quadratic sequence has a constant second difference (the differences of the differences). To find its $n$nth term: halve the second difference to get the coefficient of $n^{2}$n2, subtract that $n^{2}$n2 part from each term, then find the linear $n$nth term of what remains. The result always has the form $an^{2} + bn + c$an2+bn+c. Setting your work out in a difference table, as below, keeps every step visible and makes errors easy to spot.

Worked Example 6

Find the $n$nth term of $3, 8, 15, 24, 35, \dots$3,8,15,24,35,…

First differences: $5, 7, 9, 11$5,7,9,11 (not constant). Second differences: $2, 2, 2$2,2,2 (constant), so the $n^{2}$n2 coefficient is $\dfrac{2}{2} = 1$22​=1. Subtract $n^{2}$n2 from each term:

$n$n	$1$1	$2$2	$3$3	$4$4	$5$5
Term	$3$3	$8$8	$15$15	$24$24	$35$35
$n^{2}$n2	$1$1	$4$4	$9$9	$16$16	$25$25
Residual	$2$2	$4$4	$6$6	$8$8	$10$10

The residual $2, 4, 6, 8, 10$2,4,6,8,10 has $n$nth term $2n$2n. So the $n$nth term is $n^{2} + 2n$n2+2n.

Answer: $n^{2} + 2n$n2+2n

Verify at $n = 4$n=4: $16 + 8 = 24$16+8=24 ✓.

Worked Example 7 [Higher]

Still Higher tier. Find the $n$nth term of $4, 7, 12, 19, 28, \dots$4,7,12,19,28,…

First differences: $3, 5, 7, 9$3,5,7,9. Second differences: $2, 2, 2$2,2,2, so the $n^{2}$n2 coefficient is $1$1. Subtract $n^{2}$n2: $3, 3, 3, 3, 3$3,3,3,3,3 — a constant $3$3. So the $n$nth term is $n^{2} + 3$n2+3.

Answer: $n^{2} + 3$n2+3

Common error: forgetting to halve the second difference, and using $2n^{2}$2n2 instead of $n^{2}$n2.

Special Sequences

You should recognise several standard sequences on sight, because spotting one immediately tells you the rule without any calculation. These appear regularly on OCR papers, sometimes disguised inside a worded problem.

Sequence	First few terms	Description
Square numbers	$1, 4, 9, 16, 25$1,4,9,16,25	$n^{2}$n2
Cube numbers	$1, 8, 27, 64$1,8,27,64	$n^{3}$n3
Triangular numbers	$1, 3, 6, 10, 15$1,3,6,10,15	$\dfrac{n(n+1)}{2}$2n(n+1)​
Fibonacci-type	$1, 1, 2, 3, 5, 8$1,1,2,3,5,8	each term is the sum of the previous two
Geometric	$2, 6, 18, 54$2,6,18,54	each term is the previous one times a constant ratio

Worked Example 8

A Fibonacci-type sequence begins $2, 5, 7, 12, \dots$2,5,7,12,… Write the next two terms.

Each term is the sum of the previous two: $7 + 12 = 19$7+12=19, then $12 + 19 = 31$12+19=31.

Answer: $19$19 and $31$31

Extended Worked Examples

Worked Example 9 — Which position has a given value

The $n$nth term of a sequence is $3n + 4$3n+4. Which term has the value $61$61?

Set $3n + 4 = 61$3n+4=61: $3n = 57$3n=57, so $n = 19$n=19.

Answer: $61$61 is the $19$19th term.

Worked Example 10 — Sequence from a pattern

A pattern of tiles uses $5$5 tiles for shape $1$1, $8$8 for shape $2$2, $11$11 for shape $3$3, and so on. How many tiles are needed for shape $20$20?