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Math Help - Variation problem

  1. #1
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    Variation problem

    The sum of the first n natural numbers is equal to the sum of two quantities, the first of which is proportional to n and the second to n^2. Work out the sums of the first three and four natural numbers and hence find the formula for the sum of the first  n natural numbers.

    How do I do this?
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  2. #2
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    Quote Originally Posted by jgv115 View Post
    The sum of the first n natural numbers is equal to the sum of two quantities, the first of which is proportional to n and the second to n^2. Work out the sums of the first three and four natural numbers and hence find the formula for the sum of the first  n natural numbers.
    Can you follow this:

    n = 4; sum 1 to n = 1+2+3+4 = 10
    kn^2 + kn = 10
    16k + 4k = 10
    k = 1/2

    (1/2)n^2 + (1/2)n = 10
    (n^2 + n) / 2 = 10
    n(n+1) / 2 = 10 = sum
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