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.


Leave a Reply

Your email address will not be published. Required fields are marked *

You may use these HTML tags and attributes:

<a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <s> <strike> <strong>

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Mr Mathematics Blog

Sharing an Amount to a Given Ratio

There are three common methods for sharing an amount to a given ratio.  Depending on the age group and ability range I am teaching I would choose one approach over the other two. The three methods are: Using fractions Unitary method Using a table In this blog I will demonstrate each of the three methods […]

Plotting Scatter Graphs and Understanding Correlation

To introduce plotting scatter graphs and understanding correlation I ask students to think about the relationships between different variables and to describe how they might be related. Here’s my starter activity which students discuss in pairs then present to me on mini-whiteboards. When the students have had time to discuss the matching pairs we talk […]

Plotting Quadratic Graphs on Cartesian Axes

In recent examiner reports it is noted how important it is for students to understand the properties of a parabola  when plotting quadratic graphs on Cartesian axes.  Students who have a secure understanding of parabolas can use them to correct miscalculated values in their table of results and are more likely to attain full marks […]