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  1. 057Logical test style: what number comes next in 2, 6, 12, 20, 30, and why?Logic and brainteasersWarm upConsulting style brainteasersSales and trading

    Try it first

    What comes next?

    Show the worked solution

    42. The gaps between terms are 4, 6, 8 and 10, rising by 2 each time, so the next gap is 12 and 30 + 12 = 42. The rule underneath is that the nth term is n x (n + 1): 1 x 2, 2 x 3, 3 x 4, 4 x 5, 5 x 6, and next 6 x 7, which is 42.

    What should you do first with any number series?

    Imagine a taxi meter that charges a little more for each extra kilometre than it did for the one before. The fares look irregular, but the jumps between them tell the story. Write the differences between terms underneath the series before guessing, and if they are not constant, take differences again. Here the first differences are 4, 6, 8 and 10, and the second differences are a steady 2, which tells you the rule is a quadratic: something times something.

    Write the differences before guessing: they grow by 2 each stepSeriesFirst differencesSecond differencesn x (n + 1)21 x 262 x 3123 x 4204 x 5305 x 6426 x 7next+4+6+8+10+12+2+2+2+2a constant second difference means the rule is a square
    The differences between 2, 6, 12, 20 and 30 are 4, 6, 8 and 10, rising by a steady 2, so the next difference is 12 and the next term is 42, which is also 6 x 7 under the rule n x (n + 1).

    How do you check the answer, not just find it?

    A second route that lands on the same number is your proof. Each term splits into two neighbouring whole numbers: 2 = 1 x 2, 6 = 2 x 3, 12 = 3 x 4, 20 = 4 x 5, 30 = 5 x 6. Two methods agreeing, differences and a formula, turn a guess into an answer. The formula also lets you jump ahead: the 10th term is 10 x 11 = 110 without writing out the terms in between.

    The relationship
    an=n(n+1)a6=6×7=42a_n = n(n+1) \qquad a_6 = 6 \times 7 = 42
    a_nthe nth term of the series
    nits position, starting at 1
    What it says in wordsEach term is its position multiplied by the next position.

    Why does a bank put this in an online test?

    Numerical and logical screens are timed tightly and filter large numbers of applicants before anyone reads a CV. The skill being tested is spotting structure fast, the same skill you use when a line in a model grows by a changing amount each year. A revenue line whose yearly increase itself rises by a fixed amount has exactly this shape, and seeing it saves you from projecting the last increase forward as if it were fixed.

    Where candidates lose it

    The fast wrong answer is 40, from assuming the last gap of 10 repeats. Under time pressure candidates spot the first differences and stop before noticing that the differences are themselves growing.

    The other loss is spending a minute hunting for an exotic rule. Differences first, second differences next, then neighbouring products: those three checks crack most test series inside the time allowed.

    What the interviewer asks next

    • What is the 20th term of the series?
    • What comes next in 1, 3, 6, 10, 15, and how is that series related to this one?
    • What is the sum of the first five terms, and is there a shortcut?