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MathidiaMathematics Academy
Sequences and patterns
Lesson 43 of 75
Standard practice

Position-to-term rules

Objective. Use simple nth-term rules to generate sequence terms.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

A position-to-term rule, often called the nth-term rule, gives a formula for finding any term directly from its position number, without working through the sequence from the start. For example, the rule 4n+1 means the first term (n=1) is 5, the second term (n=2) is 9, and so on. This is more efficient than a term-to-term rule for finding a distant term, like the 50th term, since you simply substitute n=50 directly. Using this rule for 4n+1 at n=50 gives 4×50+1=201.

Worked example

A sequence has the position-to-term rule 3n − 2. Find the 8th term of the sequence.

3 … 2
Illustrated card showing 3 … 2
  1. 1Identify the position required: n=8.
  2. 2Substitute into the rule: 3×8−2.
  3. 3Calculate the multiplication first: 3×8=24.
  4. 4Subtract: 24−2=22.

Answer: 22

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Practice

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Common mistake. Pupils sometimes substitute the term's value instead of its position into the rule, confusing which number represents n. Always identify clearly which number is the position before substituting.

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