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Math Help - Fractions sum

  1. #1
    Newbie
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    Fractions sum

    Good Day!

    Please help me solve this problem!

    Find the sum of


    (1 - 1/6) + (1/2 - 1/7) + (1/3 - 1/8) + ....+ (1/995 -1/1000)


    I guess it has something to do with arithmetic progression but I can't find their common difference. I guess my idea is wrong.


    Thanks,
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  2. #2
    MHF Contributor red_dog's Avatar
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    Medgidia, Romania
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    The sum can be written as

    \sum_{k=1}^{995}\left(\frac{1}{k}-\frac{1}{k+5}\right)=

    =\sum_{k=1}^{995}\frac{1}{k}-\sum_{k=1}^{995}\frac{1}{k+5}=

    =\sum_{k=1}^{995}\frac{1}{k}-\sum_{k=6}^{1000}\frac{1}{k}=

    =\sum_{k=1}^5\frac{1}{k}+\sum_{k=6}^{995}\frac{1}{  k}-\sum_{k=6}^{995}\frac{1}{k}-\sum_{k=996}^{1000}\frac{1}{k}=

    =\sum_{k=1}^5\frac{1}{k}-\sum_{k=996}^{1000}\frac{1}{k}=

    =1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{996}-\frac{1}{997}-\frac{1}{998}-\frac{1}{999}-\frac{1}{1000}
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  3. #3
    Senior Member pacman's Avatar
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    As given by red_dog, we calculate;

    S = (1/1 - 1/6) + (1/2 - 1/7) + (1/3 - 1/8) + ....+ (1/995 -1/1,000)

    S = 1/1 + 1/2 + 1/3 + 1/4 + 1/5 - 1/996 - 1/997 - 1/998 - 1/999 - 1/1,000

    S = 1/1 + 1/2 + 1/3 + 1/4 + 1/50 - 1/996 - 1/997 - 1/998 - 1/999 - 1/1,000

    S = 137/60 - 206671037501/41251456251000

    S = 93984154068949/41251456251000

    S = 2 + 11481241566949/41251456251000

    S = 2 + 0.2783233032329780 . . . .

    S = 2.2783233032329780 . . . .
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