# Arithmetic and Geometric Sequences

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

#### 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 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.

#### Common Misconceptions

• 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 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.

## Arithmetic and Geometric Sequences Resources

### Mr Mathematics Blog

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