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  1. MHB Craftsman

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    #1
    Hello all,

    In the attached picture there is an equation. I need to fill the general expression on the left hand side, and to prove by induction that the sum is equal to the expression in the right hand side.

    I am not sure how to find the general expression. Can you kindly assist ?

    Thank you !


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  3. MHB Journeyman
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    #2
    $\dfrac{(-1)^{n+1}(2n+1)}{n^2+n}$

  4. MHB Master
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    #3
    The numerators are clearly the odd numbers so: 2n+ 1.

    The denominators are a little harder! I would have used "Newton's "divided difference" formula: adding a first term of "0", the "first differences" are 2- 0= 2, 6- 2= 4, 12- 6= 6, 20- 12= 8, 30- 20= 10; the "second differences" are 4- 2= 2, 6- 4= 2, 8- 6= 2, 10- 8= 2. Those are all "2" so all further "differences" are 0. The denominators are given by the quadratic $ \displaystyle 0+ 2n+ (2/2)n(n-1)= n^2+ n$.

    Of course, since the +/- sign alternates we need -1 to a power. The first term, with n= 1, is positive so that can be either $ \displaystyle (-1)^{n+1}$ or $ \displaystyle (-1)^{n-1}$.

  5. MHB Journeyman
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    #4
    I saw sequence of denominators, $2 ,6,12,20,30,...$, as

    $(1\cdot 2), (2 \cdot 3), (3 \cdot 4), ( 4 \cdot 5),(5 \cdot 6), ... , [n \cdot (n+1)] , ...$

  6. MHB Craftsman

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    #5 Thread Author
    I tried proving this by induction using the general statement that skeeter wrote, but I couldn't do it.

    I am stuck at the n=k+1 stage...
    Last edited by Yankel; January 13th, 2020 at 17:19.

  7. MHB Journeyman
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    #6
    note $1 + \dfrac{(-1)^{n+1}}{n+1} = \dfrac{(n+1) + (-1)^{n+1}}{n+1}$



    ${\color{red}{\dfrac{3}{2} - \dfrac{5}{6} + \dfrac{7}{12} - \dfrac{9}{20} + ... + \dfrac{(-1)^{n+1}(2n+1)}{n(n+1)}}} + \dfrac{(-1)^{(n+1)+1}[2(n+1)+1]}{(n+1)[(n+1)+1]}$

    ${\color{red}\dfrac{(n+1) + (-1)^{n+1}}{n+1}} + \dfrac{(-1)^{n+2}(2n+3)}{(n+1)(n+2)}$

    $\dfrac{(n+1)(n+2) + (-1)^{n+1}(n+2)}{(n+1)(n+2)} + \dfrac{(-1)^{n+2}(2n+3)}{(n+1)(n+2)}$

    $\dfrac{(n+1)(n+2) + (-1)^{n+1}(n+2) - (-1)^{n+1}(2n+3) }{(n+1)(n+2)}$

    $\dfrac{(n+1)(n+2) + (-1)^{n+1}[(n+2) - (2n+3)] }{(n+1)(n+2)}$

    $\dfrac{(n+1)(n+2) + (-1)^{n+2}(n+1) }{(n+1)(n+2)}$

    $\dfrac{(n+1)(n+2)}{(n+1)(n+2)}+ \dfrac{(-1)^{n+2}(n+1)}{(n+1)(n+2)}$

    $ 1 + \dfrac{(-1)^{(n+1)+1}}{(n+1)+1}$

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