Arithmetic and Geometric Sequences

Students learn how to generate and describe arithmetic and geometric sequences on a  position-to-term basis.  Learning progresses from plotting and reading coordinates in the first quadrant to describing geometric sequences using the nth term.

This unit takes place in Term 4 of Year 10 and is followed by the equations of straight line graphs.

Arithmetic and Geometric Sequences Lessons

Prerequisite Knowledge
  • Use simple formulae
    • Generate and describe linear number sequences
    • Express missing number problems algebraically
  • Pupils need to be able to use symbols and letters to represent variables and unknowns in mathematical situations that they already understand, such as:
    • missing numbers, lengths, coordinates and angles
    • formulae in mathematics and science
    • equivalent expressions (for example, a + b = b + a)
    • generalisations of number patterns

Success Criteria
  • Generate terms of a sequence from either a term-to-term or a position-to-term rule
  • Recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions ( r<sup<n where n is an integer, and r is a rational number > 0 or a surd) and other sequences
  • Deduce expressions to calculate the nth term of linear and quadratic sequences

Key Concepts
  • The nth term represents a formula to calculate any term a sequence given its position.
  • To describe a sequence it is important to consider the differences between each term as this determines the type of pattern.
  • Quadratic sequences have a constant second difference. Linear sequences have a constant first difference.
  • Geometric sequences share common multiplying factor rather than common difference.
  • Whereas a geometric and arithmetic sequence depends on the position of the number in the sequence a recurrence relation depends on the preceding terms.

Common Misconceptions
  • Students often show a lack of understanding for what ‘n’ represents.
  • A sequence such as 1, 4, 7, 10 is often described as n + 3 rather than 3n – 2.
  • Quadratic sequences can involve a linear as well as a quadratic component.
  • Calculating the product of negative numbers when producing a table of results can lead to difficulty.
  • The nth term for a geometric sequence is in the form arn-1 rather than arn.
  • Students often struggle understanding the notation of recurrence sequences. In particular, using difference values of n for a given term.


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