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Suppose we have a sequence

(1), (2, 3), (4, 5, 6). (7, 8, 9, 10), (11, 12, 13, 14, 15), ...

where (1) is the first element of the sequence, (2, 3) is the second, etc. What is the first number in the 100th element of the sequence? Be sure to do more than answer the problem. Could you use this problem with your students? What grade level would be appropriate? Could you adapt the problem to make it appropriate for at least four different grade levels?

  • Math -

    the sum of the 1st n elements is
    it will be a cubic, since it's the sum of n(n+1)/2 for n terms

    Sum of 1st n triangular numbers is


    sum of 1st 99 elements is thus (99)(100)(101)/6 = 166650

    So, the 100th element begins with the number 166651

    since this involves proof by induction, I'd say algebra 2 would be a good place to present it.

  • Math -

    Thanks Steve
    What do you mean by its a cubic?

  • Math -

    hmm. since we're talking about polynomials here, I figured it'd be clear that I meant a 3rd-degree polynomial. As opposed to a quadratic or linear.

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