Arithmetic and Geometric Sequences

Scheme of work: GCSE Higher: Year 10: Term 4: Arithmetic and Geometric Sequences

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

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

The nth term represents a formula to calculate any term in 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.

Students often show a lack of understanding of what the nth term 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.