Rukasha Maths

GCSE maths Beta

Linear sequences · Revision guide

Foundation · Algebra · 1 page in the PDF

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Learning objectives

  • Find and use a linear sequence rule.
  • Decide whether a value belongs to a sequence.

Key ideas

A linear sequence has a constant first difference.

For first term and common difference, use this position-to-term rule.

un=dn+(a−d)(first term a, difference d)u_n=dn+(a-d)\quad\text{(first term }a\text{, difference }d\text{)}

Term positions start at one unless the question states otherwise.

Check a proposed rule using the first few positions.

A sequence member must correspond to a positive whole-number position.

Worked examples

Example 1

Find the position-to-term rule and the twentieth term.

5,9,13,17,…:d=4,a=55,9,13,17,\ldots:\qquad d=4,\quad a=5
un=4n+(5−4)=4n+1u_n=4n+(5-4)=4n+1
u20=4(20)+1=81u_{20}=4(20)+1=81
un=4n+1,u20=81u_n=4n+1,\qquad u_{20}=81

Example 2

Test whether the proposed value is a sequence member.

6n+2=46⟹n=446=2236n+2=46\quad\Longrightarrow\quad n=\frac{44}{6}=\frac{22}{3}

This is not a whole-number position.

46 is not in the sequence46\ \text{is not in the sequence}

Watch out

The common difference is only the coefficient of the position. Check the constant adjustment using the first term.