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## National STEM Learning Centre

Skip to 0 minutes and 8 seconds MICHAEL ANDERSON: Arithmetic sequences in which the common difference is always the same is where each sequence goes up or down by the same amount each time. The simplest arithmetic sequences are the times tables. So the 2 times table is 2, 4, 6, 8, 10, and so on. The term-to-term rule can be written as this. The first term is 2, and to get the next term we just add 2 to the previous term. The position-to-term rule can be written as Rn equals 2n. We multiply the position by 2.

Skip to 0 minutes and 47 seconds Have a go for writing down the term-the-term rule and the position-to-term rule for the 6 times table.

Skip to 0 minutes and 56 seconds All arithmetic sequences are based on the times tables. For the sequence P, P equals 5, 9, 13, 17, 21, and so on. And the question we need to ask is, is there a common difference? In this case, yes. The common difference is 4. Each term is 4 more than the previous term. This sequence is therefore based on the 4 times table. The term-to-term rule is, the first term, P1, is 5, and to get the next term, we add 4 to the previous term. We know that the sequence is based on the 4 times table. The sequence is always 1 more than the 4 times table. Therefore, the position-to-term rule can be written as Pn is equal to 4n plus 1.

Skip to 1 minute and 49 seconds Let’s consider the sequence Q. Q equals 4, 10, 16, 22, 28, and so on. The question we need to ask is, is there a common difference? In this case, yes. The common difference is 6. Each term is 6 more than the previous term. The sequence is therefore based on the 6 times table. The term-to-term rule can be defined as, Q1, the first term, is 4, and to get the next term, we add 6 to the previous term. Let’s try and find the position-to-term rule. We know that the sequence is based on the 6 times table. The sequence is always 2 less than the 6 times table. Therefore, the position-to-term rule is Qn is equal to 6n take away 2.

# Arithmetic sequences

Arithmetic sequences are the most common type of sequence students meet. It is important to emphasise that there are two ways to describe arithmetic sequence: the term to term rule and the position to term rule. Refer back to earlier in this week if you need to clarify these definitions.

Often text books ask for the nth term of the sequence without specifying that it is the position to term rule that is required. Arithmetic sequences are sometimes referred to as ‘linear’ sequences because if we plot the terms of an arithmetic sequence on a graph the points lie on a straight line. Arithmetic sequences form a good foundation for exploring linear graphs.

In this video Michael shows that the simplest arithmetic sequences are the times tables Michael explains how to generate the term to term rules for arithmetic sequences and how the times tables can be used to help generate the position to term rule.

## Problem worksheet

Now complete question 4 from this week’s worksheet.

## Teaching resource

This SMILE resource contains three packs of games, investigations, worksheets and practical activities supporting the teaching and learning of sequences, from finding the next two terms of a simple linear sequence to exploring the limits of sequences