Level 5: Arithmetic sequences

Patterns and sequences · level 5 of 8 · rough age 12-13 (a guide for parents; anyone can start anywhere)

In level 4 you wrote a position rule like \(t_n = 3n + 2\) for patterns that grow by the same amount each time. In this level those patterns get their proper name, arithmetic sequences, and a formula that works for every one of them. You learn to rebuild a sequence from any two of its terms, to add up a hundred terms in one line (Gauss's trick), and to use differences to recognise patterns that are not arithmetic at all, such as the square numbers, and find their rules too. The level ends with arithmetic sequences at work: theatre seats, savings plans and stacks of cans.

Lesson	You will be able to
Arithmetic sequences	Recognise an arithmetic sequence, find its common difference, and use \(t_n = a + (n - 1)d\), knowing why it is \(n - 1\)
Working backwards	Find \(a\) and \(d\) from any two terms, decide whether a number belongs to a sequence, and fill in missing terms
Gauss's trick: adding an arithmetic sequence	Add \(1 + 2 + \cdots + 100\) by pairing the ends, and add any arithmetic sequence with two formulas
Finite differences	Use first, second and third differences to tell linear, quadratic and cubic patterns apart, and predict the next terms
Quadratic patterns	Find the rule \(an^2 + bn + c\) for a sequence with a constant second difference, and see it in dot pictures
Arithmetic sequences in real life	Model seats, savings, salaries and stacks with arithmetic sequences, and avoid the fence-post trap

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