Find the nth term of a linear sequence

Term-to-term rules and the nth term of a sequence

A sequence is an ordered list of terms. A term-to-term rule describes how to obtain the next term from the current one, such as add 5. A position-to-term rule gives the value at any chosen position without generating every earlier term. The nth term is a compact position-to-term rule because n stands for the term number: n = 1 means the first term, n = 20 means the twentieth term, and so on.

This distinction matters when a question asks for a distant value. Starting at 2 and adding 5 repeatedly will eventually reach the fiftieth term, but substituting n = 50 into a correct nth-term rule is faster and less prone to counting errors. Year 7 pupils should be able to use a supplied rule to generate terms and begin finding rules for arithmetic sequences.

How to find the nth term from the common difference

For the sequence 3, 7, 11, 15, the common difference is 4. Begin with 4n, whose first four values are 4, 8, 12 and 16. Each target term is one less, so the rule is 4n - 1. Check it: when n = 1, 4 × 1 - 1 = 3; when n = 3, 4 × 3 - 1 = 11. The same adjustment works at every position because both sequences increase by 4.

A useful shortcut is to calculate the zero term. Extend the arithmetic sequence backwards once by subtracting the common difference. For 3, 7, 11, 15, the zero term is -1, so the rule is 4n - 1. For 8, 13, 18, 23, the difference is 5 and the zero term is 3, giving 5n + 3. The zero term is an efficient check, not a term that normally appears in the listed sequence.

• Common difference → coefficient of n.
• Zero term → constant added to or subtracted from that multiple of n.
• Substitution → proof that the proposed rule reproduces known terms.

Decreasing sequences, decimals and using a formula

A decreasing arithmetic sequence has a negative common difference. In 20, 17, 14, 11, the difference is -3. Start with -3n, whose first value is -3, then add 23 to reach 20, so the nth term is -3n + 23. Always preserve the negative sign when using the difference as the coefficient.

The same method works with decimal or fractional differences when the sequence is arithmetic. If a rule is already supplied, substitute each required position carefully. For the rule 2n + 5, the seventh term is 2 × 7 + 5 = 19. To find a missing term, identify its position first; do not substitute the missing value itself for n.

BBC Bitesize nth term support and checked practice

The reachable BBC Bitesize Patterns and sequences topic offers a useful second explanation for pupils who searched for BBC Bitesize nth term help. The exact Oak lessons below provide the narrower method: generating terms from a position-to-term rule, finding the coefficient from the common difference, identifying the translation and checking whether a number belongs to a sequence. The downloadable Oak materials provide free worksheet and quiz practice.

Use the account-gated Twinkl walkthroughs only when access is available; they are labelled clearly rather than presented as open resources. The broad Khan Algebra Basics link and a fractional-sequences Corbett page have been removed from this activity boundary because they do not match the three Year 7 worksheet contexts closely enough. Finish practice by writing a rule, testing three known terms and using it to calculate a distant term.

Five nth-term tasks with working

• 4, 7, 10, ... has common difference 3. Start with 3n; 3n is one below each term, so the rule is 3n + 1.
• 2, 7, 12, ... has common difference 5. Start with 5n; subtract 3 to obtain the first term, so the rule is 5n − 3.
• 11, 8, 5, ... has common difference −3. Start with −3n; add 14 to obtain 11 when n = 1, so the rule is −3n + 14.
• For 4n − 5, substitute n = 1, 2 and 3: −1, 3, 7.
• Is 67 in 5n + 2? Solve 5n + 2 = 67, so 5n = 65 and n = 13. Yes, 67 is the thirteenth term.