Sequences (Nth Term)
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Revision notes for Edexcel GCSE Maths Sequences (Nth Term). Open the guide for explanations and worked examples. Written against the Edexcel GCSE Maths (1MA1) specification, so the content matches what's examinable rather than general Maths background.

Sequences (Nth Term)

What you'll learn

  • How to continue a sequence by spotting the rule.
  • How to use an nth-term formula to find terms.
  • How to decide whether a number is in a sequence.
  • How to find the nth term of an arithmetic sequence.

1. Sequences, terms and positions

A sequence is a list of numbers or patterns in order. The order matters.

Definition

Sequence words

  • A sequence is an ordered list, such as 5, 8, 12, 17.

  • A term is one number in the sequence.

  • The position tells you where the term is: first, second, third, and so on.

  • A term-to-term rule tells you how to get from one term to the next.

To continue a sequence, look at the gaps between terms.

Example

Continuing a sequence from the gaps

The first terms are 5, 8, 12, 17. Find the next two terms.

The gaps increase by 1 each time, so the next jumps are +6 and +7.

  1. Work out the gaps: from 5 to 8 is +3, from 8 to 12 is +4, and from 12 to 17 is +5.

  2. The gaps are increasing by 1 each time, so the next gaps are +6 and +7.

  3. Add 6 to 17 to get 23.

  4. Add 7 to 23 to get 30.

  5. The next two terms are 23 and 30.

Common Mistake

Assuming the gap is always the same

Do not just use the first gap. Check all the gaps you are given, because some sequences have changing gaps.

Checking a repeated add rule

If a sequence starts at a number and keeps adding the same amount, you can check whether another number appears by looking at the difference from the first term.

Example

Is a number in an add-on sequence?

The first term is 4. The rule is add 6. Is 83 a term?

Starting at 4 and adding 6 repeatedly lands on numbers congruent to 4 modulo 6, not on 83.

  1. The sequence starts 4, 10, 16, 22, and keeps adding 6.

  2. Find the difference from the first term to 83: 83 − 4 = 79.

  3. Since 79 is not a multiple of 6, you cannot reach 83 by adding 6s.

  4. So 83 is not a term in the sequence.

2. Rules that are not just adding

Some term-to-term rules tell you exactly what to do each time, such as “multiply by 2 then subtract 1”. Follow the rule carefully, one term at a time.

Example

Using a given term-to-term rule

A sequence starts with 3. The rule is: multiply the previous term by 2, then subtract 1. Find the 5th term.

Each term is made from the previous term by applying ×2 then −1 in order.

  1. Start with the first term: 3.

  2. Double 3 and subtract 1 to get the second term: 5.

  3. Double 5 and subtract 1 to get the third term: 9.

  4. Double 9 and subtract 1 to get the fourth term: 17.

  5. Double 17 and subtract 1 to get the fifth term: 33.

Fibonacci-type sequences

A Fibonacci-type sequence makes each new term by adding the two previous terms.

Example

Continuing a Fibonacci-type sequence

The sequence begins 3, 4, 7, 11, 18. Find the next two terms.

In a Fibonacci-type sequence, each new term is the sum of the two terms immediately before it.

  1. Add the last two terms: 11 + 18 = 29.

  2. The next term is 29.

  3. Add the new last two terms: 18 + 29 = 47.

  4. The next two terms are 29 and 47.

3. The nth term: using a formula

The nth term is a formula that tells you the term in position nnn.

Definition

Nth term

The letter nnn stands for the position number. So n=1n=1n=1 means the first term, n=2n=2n=2 means the second term, and so on.

The position number n tells you which term of the sequence you are looking at.

To use an nth-term formula, substitute the position number into the formula.

Example

Using a linear nth term

The nth term of a sequence is 5n+25n + 25n+2.

Substituting position numbers into 5n + 2 generates the terms of the sequence.

Find the first two terms, and decide whether 37 is a term.

  1. For the first term, use n=1n=1n=1: 5(1)+2=75(1)+2=75(1)+2=7.

  2. For the second term, use n=2n=2n=2: 5(2)+2=125(2)+2=125(2)+2=12.

  3. The first two terms are 7 and 12.

  4. To check whether 37 is a term, solve 5n+2=375n + 2 = 375n+2=37:

    5n+2=375n=35n=7\begin{aligned} 5n + 2 &= 37 \\ 5n &= 35 \\ n &= 7 \end{aligned}5n+25nn​=37=35=7​

  5. Since 7 is a positive whole-number position, 37 is a term.

Common Mistake

Starting with n = 0

In GCSE sequence questions, the first term is normally when n=1n=1n=1, not when n=0n=0n=0, unless the question says otherwise.

Formulae with squares

If the formula contains n2n^2n2, square the position number first.

Example

Using a square nth term

The nth term is n2+5n^2 + 5n2+5.

Find the first three terms, and decide whether 54 is a term.

  1. Use n=1n=1n=1: 12+5=61^2+5=612+5=6.

  2. Use n=2n=2n=2: 22+5=92^2+5=922+5=9.

  3. Use n=3n=3n=3: 32+5=143^2+5=1432+5=14.

  4. To check 54, solve n2+5=54n^2 + 5 = 54n2+5=54:

    n2+5=54n2=49n=7\begin{aligned} n^2 + 5 &= 54 \\ n^2 &= 49 \\ n &= 7 \end{aligned}n2+5n2n​=54=49=7​

  5. Since 7 is a positive whole-number position, 54 is a term.

4. Finding the nth term of an arithmetic sequence

Definition

Arithmetic sequence

An arithmetic sequence has the same difference between consecutive terms. Consecutive terms are terms next to each other. The fixed difference is called the common difference.

For an arithmetic sequence, the nth term has the form dn+cdn + cdn+c.

  • ddd is the common difference.
  • ccc is the adjustment needed to match the actual sequence.
Key Idea

The main nth-term idea

The number in front of nnn comes from the common difference, not from the first term.

Example

Finding an increasing nth term

Find the nth term of 8, 13, 18, 23, 28.

Comparing the sequence with the 5 times table shows a constant adjustment of +3.

  1. The common difference is 5, because each term increases by 5.

  2. Start with 5n5n5n. Its first few values are 5, 10, 15, 20, 25.

  3. Compare with the actual sequence: 8, 13, 18, 23, 28.

  4. Each actual term is 3 bigger than the matching 5n5n5n value.

  5. So the nth term is 5n+35n + 35n+3.

Decreasing arithmetic sequences

If the sequence goes down, the common difference is negative.

Example

Finding a decreasing nth term

Find the nth term of 26, 22, 18, 14, 10.

  1. The sequence goes down by 4 each time, so the common difference is negative 4.

  2. Start with −4n-4n−4n. Its first few values are negative 4, negative 8, negative 12.

  3. To get from negative 4 to 26, add 30.

  4. So the nth term is −4n+30-4n + 30−4n+30.

  5. Check: when n=1n=1n=1, −4n+30=26-4n+30=26−4n+30=26.

Tip

Quick check

After finding an nth term, test it with n=1n=1n=1 and n=2n=2n=2. If it does not give the first two terms, fix it before moving on.

5. Pattern sequences

Pattern questions work like number sequences. The pattern number is the position number.

Count carefully, then look for what changes each time.

Example

Counters in a growing pattern

A counter pattern has totals 6, 10, 14 for patterns 1, 2 and 3. Find a rule and work out pattern 8.

The first three counter patterns grow by adding four counters each time.

  1. The totals increase by 4 each time, so this is an arithmetic sequence.

  2. Start with 4n4n4n. The first value of 4n4n4n is 4.

  3. Pattern 1 actually has 6 counters, which is 2 more than 4.

  4. So the rule is 4n+24n + 24n+2.

  5. For pattern 8, use n=8n=8n=8: 4(8)+2=344(8)+2=344(8)+2=34.

  6. Pattern 8 has 34 counters.

Exam technique

In the exam

  1. Write down the gaps between terms before deciding on a rule.

  2. For an nth-term formula, substitute n=1n=1n=1, n=2n=2n=2, or the position the question asks for.

  3. If checking whether a number is a term, solve for nnn. It must be a positive whole number.

Self review

Check yourself

  • Can you explain the difference between a term-to-term rule and an nth-term rule?

  • If a sequence goes down by 3 each time, what sign should the number in front of nnn have?

  • How can you check that your nth-term expression is correct?

Recap questions

Test yourself with 5 quick questions on this guide. Answer them all correctly to complete it.

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