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Arithmetic Sequences

An arithmetic sequence increases (or decreases) by the same amount each step — the common difference. A single formula gives any term you want without listing them all.

25 min Video by Zeeshan Zamurred Sequences and Series

Edexcel A level Maths: 3.1 Arithmetic SequencesWatch the full walkthrough before the notes below.

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What you'll be able to do

Identify an arithmetic sequence and its common difference
Use the nth term formula
Find a missing term or position
Set up equations from given terms

Common difference

In an arithmetic sequence each term differs from the last by a constant $common difference$ $d$ . The first term is $a$ .

d = u_{n} - u_{n - 1}

The fixed step between consecutive terms.

The nth term

Any term is the first term plus $(n - 1)$ steps of $d$ .

u_{n} = a + (n - 1) d

a

= first term,

d

= common difference.

a = 3

d = 4

u_{10} = 3 + 9 (4)

Answer

39

Setting up equations

Given two terms, write each with the nth-term formula to get simultaneous equations in $a$ and $d$ , then solve.

a + 2 d = 11

and

a + 6 d = 27

2Subtract:

4 d = 16 \Rightarrow d = 4

a = 3

Answer

a = 3, d = 4

Tip — Two given terms → two equations of the form a + (n−1)d. Subtract to find d first.

Formula recap

u_{n} = a + (n - 1) d

nth term.

d = u_{n} - u_{n - 1}

Common difference.

Common mistakes to avoid

Using nd instead of (n − 1)d.

The first term already counts; add (n − 1) differences.

Mixing up a and d.

a is the first term; d is the step between terms.

Key takeaways

Arithmetic: constant common difference d.
nth term: uₙ = a + (n − 1)d.
Two given terms give simultaneous equations for a and d.

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