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Welcome to GCSE Edexcel Maths revision.

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Topic M 11: Sequences.

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This video covers Higher tier.

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It includes the shared content and the labelled Higher extensions.

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A term-to-term rule tells you how to get the next term.

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A position-to-term rule gives a term directly from its position n, starting with n equals 1 unless stated otherwise.

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An arithmetic sequence has a constant difference.

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For 5, 8, 11, 14, and so on the difference is 3, so the nth term is 3 n plus 2: compare with 3, 6, 9, 12, and so on.

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Worked example: For 12, 8, 4, 0, and so on the nth term is 16 minus 4 n.

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The tenth term is 16 minus 40 equals minus 24.

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The difference can be negative.

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To test whether a value belongs to a sequence, set its nth-term expression equal to that value.

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For 3 n plus 2 equals 50, n equals 16, so 50 is a term; n must be a positive integer.

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Square numbers are 1, 4, 9, 16, and so on; cubes are 1, 8, 27, 64, and so on; triangular numbers are 1, 3, 6, 10, and so on, increasing by 2, 3, 4, and so on.

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First and second differences

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A Fibonacci-type sequence adds the two previous terms.

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Starting 2, 3 gives 2, 3, 5, 8, 13, and so on.

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Its first two terms must be specified.

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A geometric sequence multiplies by a constant ratio.

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Starting at 3 and multiplying by 2 gives 3, 6, 12, 24, and so on.

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Starting at 8 and multiplying by 1 over 2 gives 8, 4, 2, 1, and so on.

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A quadratic sequence has constant non-zero second differences.

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Recognise it by finding first differences and then differences of those differences; the first differences are not constant.

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Worked example: For a pattern made from adjoining squares, the first square needs four sticks and each extra square adds three.

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The nth pattern needs 3 n plus 1 sticks; pattern 20 needs 61.

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A few terms can fit more than one rule.

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Use the stated pattern or enough structural information; do not claim a rule is uniquely determined by only two or three numbers.

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For an squared plus bn plus c, the constant second difference is 2 a.

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Divide it by 2 to find a, then subtract an squared from each term to leave a linear sequence.

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Worked example: 3, 8, 15, 24, and so on has differences 5, 7, 9 and second difference 2.

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Subtract n squared to get 2, 4, 6, 8, and so on, so the rule is n squared plus 2 n.

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For first term a and common ratio r, the nth term is ar to the power of open bracket n minus 1 close bracket .

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The first term uses power 0, so it equals a, not ar.

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Worked example: 5, 15, 45, and so on has nth term 5 multiplied by 3 to the power of open bracket n minus 1 close bracket .

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Its fifth term is 5 multiplied by 3 to the power of open bracket 4 close bracket equals 405.

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Higher problems can use ratios such as square root of 2; standard positive rational ratios are shared content.

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That completes Sequences.

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Revisit the notes and test yourself on the revision website.