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Math Help - prove that the sum of the integers is

  1. #1
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    Arrow prove that the sum of the integers is

    The positive integers are bracketed as follows:
    (1), (2,3), (4,5,6), . . .
    where there are r integers in the rth bracket. Prove that the sum of the integers in the rth bracket is

     \frac{r}{2} (r^2 +1)

    this is what I did
    using the formula for arithmetic sequence,
    we know that there are r integers. and the difference is 1

    so that becomes
    Sum of r integers =  \frac{r}{2}(2a + (r-1)) where a is the first term of the sequence

    However, I can't figure out what a (a.k.a the first term) is equal to


    any ideas?

    thanks
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  2. #2
    MHF Contributor

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    Paris, France
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    Quote Originally Posted by differentiate View Post
    The positive integers are bracketed as follows:
    (1), (2,3), (4,5,6), . . .
    where there are r integers in the rth bracket. Prove that the sum of the integers in the rth bracket is

     \frac{r}{2} (r^2 +1)

    this is what I did
    using the formula for arithmetic sequence,
    we know that there are r integers. and the difference is 1

    so that becomes
    Sum of r integers =  \frac{r}{2}(2a + (r-1)) where a is the first term of the sequence

    However, I can't figure out what a (a.k.a the first term) is equal to
    Note that a=(1+2+3+4+\cdots+(r-1))+1 because a is comes after the first bracket (1 term), the second brack (2 terms), etc. , and after the (r-1)-th bracket.
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  3. #3
    Member
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    ohh that makes sense now

    thanks!
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