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Math Help - Summation

  1. #1
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    Summation

    How do you calculate the summation when the top is n-1?

    For example, if I want to find the general term of the sequence \{ a_n \}1,3,7,13,21,31,...

    The first order difference is \{ b_n \} 2,4,6,8,10,...

    \{ b_n \}=2n

    \{a_n \}=a_1+ \sum_{k=1}^{n-1} b_k=\sum_{k=1}^{n-1} 2k

    How do I calculate the sum of k?
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  2. #2
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    Each term = n^2 - n + 1 ; if n=6, term=31
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  3. #3
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    {a_n}=1+\sum_{k=1}^{n-1}2k=1+2\sum_{k=1}^{n-1}k=1+2\frac{(n-1)n}{2}=1+(n-1)(n)
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  4. #4
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    Same as mine, Krahl; n=6, 1 + 6(5) = 31 ; not a sum
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  5. #5
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    So for \sum_{k=1}^{n-1} k=\frac {n(n-1)}{2}? How do you calculate that?

    How about k^2?
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  6. #6
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    What do you mean?
    The sum of the first n natural numbers is

    1+2+3+...+n=\frac{(n)(n+1)}{2}

    so the sum of the first n-1 natural numbers is

    1+2+3+...+(n-1)=(n-1)(n-1+1)=\frac{(n-1)(n)}{2}

    and the sum of the first n squared numbers is \frac{n(n+1)(2n+1)}{6}

    But these two are different formulae
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  7. #7
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    Quote Originally Posted by Krahl View Post
    What do you mean?
    The sum of the first n natural numbers is

    1+2+3+...+n=(n)(n+1)

    so the sum of the first n-1 natural numbers is

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

    and the sum of the first n squared numbers is \frac{n(n+1)(2n+1)}{6}

    But these two are different formulae
    The formula you stated for k^2was for \sum_{k=1}^n, not n-1

    EDIT: so if you want to find the sum for n-1, you just substitute n-1 as n into the regular formulas?
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  8. #8
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    yes so just replace n in that formula with n-1.Thats all you need to do. try it out see if it works

    so it becomes \frac{(n-1)(n-1+1)(2(n-1)+1)}{6}

    is that ok?
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  9. #9
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    My bad; misread the question...
    as penance, I'll recite the Hoooooly Rosary twice
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  10. #10
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    Quote Originally Posted by Krahl View Post
    yes so just replace n in that formula with n-1.Thats all you need to do. try it out see if it works

    so it becomes \frac{(n-1)(n-1+1)(2(n-1)+1)}{6}

    is that ok?
    I understood it at the same time you typed the post!

    Thank you
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