Binomial theorem

  • Telescoping series
    problem

    Telescoping series

    Age
    16 to 18
    Challenge level
    filled star filled star empty star

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

  • Tens
    problem

    Tens

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