Mathematical induction

  • Gosh Cosh
    problem

    Gosh cosh

    Age
    16 to 18
    Challenge level
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    Explore the hyperbolic functions sinh and cosh using what you know about the exponential function.
  • Placeholder: several colourful numbers
    article

    Binomial coefficients

    An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.
  • Particularly general
    problem

    Particularly general

    Age
    16 to 18
    Challenge level
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    By proving these particular identities, prove the existence of general cases.
  • Farey Fibonacci
    problem

    Farey Fibonacci

    Age
    16 to 18
    Challenge level
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    Investigate Farey sequences of ratios of Fibonacci numbers.

  • Farey Neighbours
    problem

    Farey neighbours

    Age
    16 to 18
    Challenge level
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    Farey sequences are lists of fractions in ascending order of magnitude. Can you prove that in every Farey sequence there is a special relationship between Farey neighbours?
  • Tens
    problem

    Tens

    Age
    16 to 18
    Challenge level
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    When is $7^n + 3^n$ a multiple of 10? Can you prove the result by two different methods?
  • Elevens
    problem

    Elevens

    Age
    16 to 18
    Challenge level
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    Add powers of 3 and powers of 7 and get multiples of 11.
  • Fibonacci Fashion
    problem

    Fibonacci fashion

    Age
    16 to 18
    Challenge level
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    What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?
  • Golden Fractions
    problem

    Golden fractions

    Age
    16 to 18
    Challenge level
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    Find the link between a sequence of continued fractions and the ratio of succesive Fibonacci numbers.
  • Water Pistols
    problem

    Water pistols

    Age
    16 to 18
    Challenge level
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    With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?