Rukasha Maths

GCSE maths Beta

Quadratic sequences · Revision guide

Higher · Algebra · 1 page in the PDF

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Learning objectives

  • Find a quadratic rule from first and second differences.
  • Use the rule to calculate a term at an integer position.

Key ideas

First differences compare consecutive terms; second differences compare consecutive first differences.

A quadratic sequence has a constant second difference twice its quadratic coefficient.

Tn=an2+bn+c,second difference=2aT_n=an^2+bn+c,\qquad\text{second difference}=2a

Subtract the square part; find a linear rule for the remainder.

The first term is position one unless another start is specified.

n∈{1,2,3,…}n\in\{1,2,3,\ldots\}

Worked examples

Example 1

Find the rule for the sequence.

3,8,15,24,35,…3,8,15,24,35,\ldots
first differences:5,7,9,11;second differences:2,2,2\text{first differences}:5,7,9,11;\quad\text{second differences}:2,2,2

Subtract the square-number sequence to obtain a linear remainder.

a=22=1;remainder:2,4,6,8,10,…a=\frac22=1;\qquad\text{remainder}:2,4,6,8,10,\ldots
Tn−n2=2nT_n-n^2=2n
Tn=n2+2nT_n=n^2+2n

Example 2

Find the fourth term of the quadratic sequence.

Tn=2n2−n,T4=2(42)−4T_n=2n^2-n,\qquad T_4=2(4^2)-4
T4=28T_4=28

Watch out

The quadratic coefficient is half the second difference. Check the complete rule against several original terms.