Arithmetic and Geometric Sequences

Generating a Sequence

Linear Nth Term

Quadratic Nth Term

Geometric Sequences

Recurrence Formulae

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 ar^n-1 rather than arⁿ.
• Students often struggle understanding the notation of recurrence sequences. In particular, using difference values of n for a given term.