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Algebra · Subtopic 6 of 9

Sequences

A sequence follows a rule. Linear sequences go up by a constant difference, so their nth term is $dn+c$ . Quadratic sequences have a constant second difference; geometric sequences multiply by the same number each time; Fibonacci-type sequences add the previous two terms.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Use term-to-term and position-to-term rules.
Find and use the nth term of a linear sequence.
Find the nth term of a simple quadratic sequence.
Recognise square, cube, triangular, geometric and Fibonacci-type sequences.

Pause and think

The sequence $7,\ 11,\ 15,\ 19,\ldots$ goes up by $4$ . Without listing every term, how could you find the $100$ th term?

Section 2 of 14

Core idea

Start hereThe 100th pattern

Matchstick squares in a row: 4, 7, 10, 13 matches... How many for 100 squares? Drawing it would take ages. A formula for the $n$ th term jumps straight there.

The big idea

Linear: constant difference $d$ , $n$ th term $dn+c$ . Quadratic: constant second difference, $n$ th term $an^2+bn+c$ with $a=\frac{\text{second difference}}2$ .

1. Linear

The sequence 5, 8, 11, 14, … plotted: a straight line of points

Gap 3, so it is the 3 times table shifted:

3n+2

. Check

n=1

5

✓.

Why

dn+c

$u_n=u_1+(n-1)d$
Start at $u_1$ , add $d$ $(n-1)$ times.
$=dn+(u_1-d)$
Rearrange: $c=u_1-d$ (the "zeroth term").

Worked example $5,8,11,14,\ldots$

$n$ th term; is 100 a term?

Goal: Formula; yes/no.
Strategy: $d=3$ ; zeroth term $5-3=2$ .

$u_n=3n+2$
$d=3$ , $c=2$ .
$3n+2=100\Rightarrow n=32.67$
Not a whole number.

Answer

3n+2

; 100 is not a term.

✓ Check

n=4

14

✓.

Pattern sequences: matchstick squares

Each new square adds 3 sticks.

2. Quadratic

Worked example $2,5,10,17,26$

$n$ th term.

Goal: Formula.
Strategy: Second difference → $a$ ; subtract $an^2$ ; what's left is linear.

$3,5,7,9;\quad2,2,2$
First and second differences.
$a=\tfrac22=1$
Half the second difference.
$u_n-n^2=1,1,1,\ldots$
Leftover is constant.

Answer

n^2+1

✓ Check

n=5

26

✓.

Common misconception $a$ = second difference It's half:

an^2

has second difference

2a

Pause and think

$n$ th term of $7,4,1,-2$ ?

Summary

Linear: $dn+c$ .
Quadratic: half the second difference.

nth term: Formula for any term.

HistoryFibonacci's rabbits. In Liber Abaci (1202), Leonardo of Pisa, known as Fibonacci, asked how many pairs of rabbits there would be after a year if each pair produced a new pair every month. The answer follows

1,1,2,3,5,8,13,\ldots

. The same book brought Hindu-Arabic numerals to Europe.

Fun factSunflowers count in Fibonacci. The seeds in a sunflower head spiral in two directions, and the numbers of spirals are usually neighbouring Fibonacci numbers, such as

34

and

55

. Pine cones and pineapples show the same pattern; it's the most efficient way to pack seeds.

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