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MathidiaMathematics Academy
Sequences and patterns
Lesson 44 of 75
Challenge

Find the nth term

Objective. Derive nth-term rules for linear sequences.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

Finding the nth term of a linear sequence means working out a formula an+b that generates every term when n is substituted with the position number. The coefficient a is the common difference between consecutive terms, found by subtracting one term from the next. The constant b is found by checking what needs to be added or subtracted to the first multiple of a to reach the first term. For the sequence 7, 11, 15, 19, the difference is 4, so the rule starts as 4n, and since 4×1=4 needs +3 to reach 7, the nth term is 4n+3.

Worked example

Find the nth term rule for the sequence 6, 10, 14, 18, ...

6 … 10
Illustrated card showing 6 … 10
  1. 1Find the common difference: 10−6=4.
  2. 2Start the rule with 4n.
  3. 3Compare 4×1=4 to the first term 6: the constant is 6−4=2.
  4. 4Check with n=3: 4×3+2=14, which matches the third term.

Answer: 4n + 2

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Common mistake. Pupils often use the first term itself as the constant, writing 4n+7 for the sequence 7, 11, 15, 19 instead of correctly finding 4n+3. Always check the constant by comparing to 4×1 for the first term.

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