Binomial theorem

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    article

    Binomial coefficients

    An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.
  • 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?
  • Bina-ring
    problem

    Bina-ring

    Age
    16 to 18
    Challenge level
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    Investigate powers of numbers of the form (1 + sqrt 2).
  • 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.
  • Discrete Trends
    problem

    Discrete trends

    Age
    16 to 18
    Challenge level
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    Find the maximum value of n to the power 1/n and prove that it is a maximum.
  • Summit
    problem

    Summit

    Age
    16 to 18
    Challenge level
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    Prove that the sum from t=0 to m of (-1)^t/t!(m-t)! is zero.
  • Remainder Hunt
    problem

    Remainder hunt

    Age
    16 to 18
    Challenge level
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    What are the possible remainders when the 100-th power of an integer is divided by 125?
  • Binomial
    problem

    Binomial

    Age
    16 to 18
    Challenge level
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    By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
  • Growing
    problem

    Growing

    Age
    16 to 18
    Challenge level
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    Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)
  • Telescoping series
    problem

    Telescoping series

    Age
    16 to 18
    Challenge level
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    Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.