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Math Help - Summation proof

  1. #1
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    Summation proof

    I have conjectured that

    \sum^{n}_{i=m}i = \sum^{n}_{i=m}( n+m-i).

    How would one go about constructing a proof for this?

    Cheers.
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  2. #2
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    Re: Summation proof

    Split the right hand side into

    \sum^{n}_{i=m}(n+m)- \sum^{n}_{i=m}(i)

    And note that \sum^{n}_{i=m}(i)= \sum^{n}_{i=1}(i)-\sum^{m}_{i=1}(i)
    You should know how to sum integers from 1 to n
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  3. #3
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    Re: Summation proof

    Got it out after I realised you made a typo

    \displaystyle\begin{align*}\sum^{n}_{i=m}{(n+m-i)} &= \sum^{n}_{i=m}{n} + \sum^{n}_{i=m}{m} - \sum^{n}_{i=m}i \\&= \sum^{n}_{i=1}n - \sum^{m-1}_{i=1}n + \sum^{n}_{i=1}m - \sum^{m-1}_{i=1}m - \sum^{n}_{i=m}i \\&= n^2 - (m-1)n + nm - (m-1)m -\sum^{n}_{i=m}{i} \\&= n^2 + n - m^2 + m -\left[ \frac{n(n+1)}{2} - \frac{(m-1)m}{2} \right] \\& = 2 \left( \frac{n(n+1)}{2} - \frac{(m-1)m}{2} \right) - \left( \frac{n(n+1)}{2} - \frac{(m-1)m}{2}\right) \\&= \frac{n(n+1)}{2} - \frac{m(m+1)}{2} \\&= \sum^{n}_{i=1}{i} - \sum^{m-1}_{i=1}{i} \\&= \sum^{n}_{i=m}{i}  \end{align*}
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