Algebra - Sequences: recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions (rⁿ where n is an integer, and r is a rational number > 0 or a surd) and other sequences (Higher tier)

The idea

Higher tier only. You must know all the families: triangular 1, 3, 6, 10, 15, 21; squares 1, 4, 9, 16, 25; cubes 1, 8, 27, 64, 125; arithmetic progressions such as 7, 11, 15, 19 with common difference 4; Fibonacci type sequences such as 2, 5, 7, 12, 19, where each term adds the two before it, giving 31 next; quadratic sequences such as 2, 5, 10, 17, with first differences 3, 5, 7 and constant second difference 2; and geometric progressions with rational ratio, such as 16, 8, 4, 2 with ratio ½. Higher tier adds surd ratios: 1, √2, 2, 2√2, 4 has ratio √2 and rule (√2)ⁿ⁻¹. 'Other sequences' means any rule you can state, such as 1/2, 2/3, 3/4, 4/5 with nth term n/(n + 1), creeping towards 1. Work from structure: say what turns one term into the next, or position into term, then test the rule on a term you were given.

What earns the marks

• The common ratio, including when it is a surd, such as dividing 2√2 by 2 to get √2.
• A term left in exact surd form rather than rounded to a decimal.
• Stating the rule of an unfamiliar sequence, in words or in algebra, even if a later term is wrong.
• Testing a proposed rule against a term you were given, which supports the rule but does not by itself prove it holds for every term.
• Recognising and continuing the standard families, including triangular, square, cube, arithmetic, Fibonacci and quadratic sequences.

Common mistakes and how to put them right

• Rounding √2 to 1.41 part way through, so the later terms drift and no longer match the sequence.

• Squaring the ratio when only one step is taken, so multiplying by 2 where the step is a multiplication by √2.

• Assuming an unfamiliar sequence must be quadratic and forcing second differences that are not actually constant.

• Treating a sequence of fractions as though only the numerators matter, and missing that the denominators follow their own rule.

Exam tips

• Keep surds exact the whole way through; a term such as 8√3 is the form the mark scheme wants, not a rounded 13.9.
• If neither the differences nor the ratios are constant, look at the terms as fractions or as powers before you decide there is no rule.
• Say which family a sequence belongs to in words before you calculate, because the reason often carries a mark of its own.
• Check the standard families first: if the gaps are constant it is arithmetic, if the second differences are constant it is quadratic, and if the ratios are constant it is geometric.