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Math Help - Partial Sum of a Sequence

  1. #1
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    Partial Sum of a Sequence

    This is an exercise in my book that I can't figure out. And there's no answer sheet for it so I don't know if I did it right.

    an = 1/n+1 - 1/n+2

    I got these as the first terms of the sequence:
    a1 = 1/6
    a2 = 1/12
    a3 = 1/20
    a4 = 1/30

    Based on those, I got these as the first partial sums of the sequence:
    S1 = 1/6
    S2 = 1/4
    S3 = 3/10
    S4 = 1/3

    I feel the partial sums are wrong, and I can't figure out the Nth partial sum of the sequence (Sn = ?).

    Any help or pointer would be greatly appreciated.

    Thank you.
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  2. #2
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    Re: Partial Sum of a Sequence

    Quote Originally Posted by RobinC View Post
    This is an exercise in my book that I can't figure out. And there's no answer sheet for it so I don't know if I did it right.

    an = 1/n+1 - 1/n+2

    I got these as the first terms of the sequence:
    a1 = 1/6
    a2 = 1/12
    a3 = 1/20
    a4 = 1/30

    Based on those, I got these as the first partial sums of the sequence:
    S1 = 1/6
    S2 = 1/4
    S3 = 3/10
    S4 = 1/3

    I feel the partial sums are wrong, and I can't figure out the Nth partial sum of the sequence (Sn = ?).
    I am not at all sure what you are doing there.
    Is it S_N  = \sum\limits_{n = 1}^N {\left( {\frac{1}{{n + 1}} - \frac{1}{{n + 2}}} \right)} ~?
    If it is then
    S_1=\frac{1}{2}-\frac{1}{3}
    S_2=\frac{1}{2}-\frac{1}{4}
    S_3=\frac{1}{2}-\frac{1}{5}
    S_4=\frac{1}{2}-\frac{1}{6}
    ~ \vdots ~
    so on.
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  3. #3
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    Re: Partial Sum of a Sequence

    Quote Originally Posted by Plato View Post
    I am not at all sure what you are doing there.
    Yes, I'm that confused

    The question states: Find the four partial sums and the nth partial sum of the sequence an.

    All I'm given is: a_n = {\left {\frac{1}{{n + 1}} - \frac{1}{{n + 2}}} \right}

    From what I read, you first find the value of a_1, a_2, a_3, a_4
    Then you find S_1, S_2, S_3, S_4
    So:
    S_1 = a_1
    S_2 = a_1 + a_2
    S_3 = a_1 + a_2 + a_3
    S_3 = a_1 + a_2 + a_3 +  a_4

    So my calculation goes like:
    S_1=\frac{1}{6}

    S_2=\frac{1}{6} + \frac{1}{16} = \frac{1}{4}

    S_3=\frac{1}{6} + \frac{1}{16} + \frac{1}{20} = \frac{3}{10}

    S_4=\frac{1}{6} + \frac{1}{16} + \frac{1}{20} + \frac{1}{30} = \frac{1}{3}

    Maybe I'm doing it wrong, but this is my first math class in over 8yrs
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  4. #4
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    Re: Partial Sum of a Sequence

    Quote Originally Posted by RobinC View Post
    Yes, I'm that confused
    The question states: Find the four partial sums and the nth partial sum of the sequence an.
    All I'm given is: a_n = {\left {\frac{1}{{n + 1}} - \frac{1}{{n + 2}}} \right}
    From what I read, you first find the value of a_1, a_2, a_3, a_4
    Then you find S_1, S_2, S_3, S_4
    You are confused
    Lets do S_4.

    \begin{align*}S_4  &= a_1  + a_2  + a_3  + a_4\\ &=\left( {\frac{1}{2} - \frac{1}{3}} \right) + \left( {\frac{1}{3} - \frac{1}{4}} \right) + \left( {\frac{1}{4} - \frac{1}{5}} \right) + \left( {\frac{1}{5} - \frac{1}{6}} \right)\\ &=\frac{1}{2}-\frac{1}{6}\end{align*}

    Removing the parentheses all but the first and last fractions collapse.
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  5. #5
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    Re: Partial Sum of a Sequence

    Ah.... okay so you don't actually input the result of a_n rather, you input the formula for it with the updated values for N.

    So instead of: S_2 = \left {\frac{1}{6} + \frac{1}{12}} \right

    We use:  S_2 = \left( {\frac{1}{2} - \frac{1}{3}} \right) + \left( {\frac{1}{3} - \frac{1}{4}} \right)

    So S_2 = \left {\frac{1}{2} - \frac{1}{4}} \right
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  6. #6
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    Re: Partial Sum of a Sequence

    Quote Originally Posted by RobinC View Post
    Ah.... okay so you don't actually input the result of a_n rather, you input the formula for it with the updated values for N.

    So instead of: S_2 = \left {\frac{1}{6} + \frac{1}{12}} \right

    We use:  S_2 = \left( {\frac{1}{2} - \frac{1}{3}} \right) + \left( {\frac{1}{3} - \frac{1}{4}} \right)

    So S_2 = \left {\frac{1}{2} - \frac{1}{4}} \right
    That is correct. These are very famous and useful series in mathematics. They are known as collapsing sums .
    Once you have the final form, then combine.

    Thus, S_n=\frac{1}{2}-\frac{1}{n+2}=\frac{n}{2n+4}
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  7. #7
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    Re: Partial Sum of a Sequence

    Thank you so much for your help! It makes a lot more sense now
    Last edited by RobinC; September 21st 2011 at 03:27 PM.
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