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Math Help - sums .

  1. #1
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    sums .

    Deduce the sum of the series .

    (1) 1^2-2^2+3^2-4^2+...+(2n-1)^2-(2n)^2+(2n+1)^2


    (2) 25^2-26^2+27^2-28^2+...+49^2-50^2


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  2. #2
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    Quote Originally Posted by thereddevils View Post
    Deduce the sum of the series .

    (1) 1^2-2^2+3^2-4^2+...+(2n-1)^2-(2n)^2+(2n+1)^2



    a^2 -b^2 =(a+b)(a-b)

    Hence
    1^2-2^2+3^2-4^2+...+(2n-1)^2-(2n)^2+(2n+1)^2

    = (1+2)(1-2) +~(2+3)(2-3) + ~......... ( ~( 2n-1) + 2n)~) ( ~( 2n-1)- 2n)~) + (2n+1)^2

     <br />
= -1[ 3 +7 +...........(4n-1) ] + (2n+1)^2<br />

     <br />
=-1[\frac{n}{2}(2\times 3 + (n-1)4)] + (2n+1)^2 <br />
............................Using AP

     <br />
= -1[ n(2n+1) ] (2n+1)^2 <br />

    = -1[2n^2 +n] + 4n^2 + 1 +4n

    = 2n^2 + 3n +1

    = (2n+1)(n+1)



    (2)
    Now for this try it the same way

     <br />
= -1[ 51 +55 +...........(4\times 25 -1) ] <br />

     <br />
=-1[\frac{25}{2}(2\times 51 + (13-1)4)] <br />

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