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3.1PureCore

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 Sequences and Series
Edexcel A level Maths: 3.1 Arithmetic SequencesCurated by Kepler, not produced by us. Watch it 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
1

Common difference

In an arithmetic sequence each term differs from the last by a constant common difference\textbf{common difference} dd. The first term is aa.

d=un−un−1d = u_{n} - u_{n-1}
The fixed step between consecutive terms.
2

The nth term

Any term is the first term plus (n−1)(n - 1) steps of dd.

un=a+(n−1)du_n = a + (n - 1)d
aa = first term, dd = common difference.
Example 1Find the 10th term of 3, 7, 11, …
1a=3a = 3, d=4d = 4.
2u10=3+9(4)u_{10} = 3 + 9(4).
Answer
3939
3

Setting up equations

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

Example 1The 3rd term is 11 and the 7th is 27. Find aa and dd.
1a+2d=11a + 2d = 11 and a+6d=27a + 6d = 27.
2Subtract: 4d=16⇒d=44d = 16 \Rightarrow d = 4, a=3a = 3.
Answer
a=3, d=4a = 3, \ d = 4

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

Formula recap

un=a+(n−1)du_n = a + (n-1)d
nth term.
d=un−un−1d = 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.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. The nth term of an arithmetic sequence is:

    • a+(n−1)da + (n-1)d
    • arn−1ar^{n-1}
    • a+nda + nd
    • an+dan + d
    Show answer and explanation

    Answer: a+(n−1)da + (n-1)d

    First term plus (n−1) steps.

  2. The common difference of 3, 7, 11, … is:

    • 44
    • 33
    • 77
    • 11
    Show answer and explanation

    Answer: 44

    Each term increases by 4.

  3. The 10th term of 3, 7, 11, … is:

    • 3939
    • 4343
    • 4040
    • 3636
    Show answer and explanation

    Answer: 3939

    3 + 9(4) = 39.

  4. If the 3rd term is 11 and 7th is 27, then d =

    • 44
    • 55
    • 1616
    • 22
    Show answer and explanation

    Answer: 44

    4d = 16.

  5. For that sequence, the first term a is:

    • 33
    • 1111
    • 55
    • 77
    Show answer and explanation

    Answer: 33

    a + 2(4) = 11.

  6. A sequence with a constant difference is called:

    • Arithmetic
    • Geometric
    • Quadratic
    • Periodic
    Show answer and explanation

    Answer: Arithmetic

    Constant common difference.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.