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Thread: subsequence

  1. #1
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    subsequence

    (an)=(2n-1) is a sequence.
    (an+1) and (a2n) are subsequences of (an).
    Is the following statement true?
    "Sum of the subsequences is also a subsequence."

    According to me this is wrong. According to the book true.
    When we put n+1 in place of n we get 2n+1
    When we put 2n in place of n we get 4n-1
    If we sum 2n+1+4n-1=6n.So the sum is not a subsequence.
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  2. #2
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    Re: subsequence

    Quote Originally Posted by kastamonu View Post
    (an)=(2n-1) is a sequence.
    (an+1) and (a2n) are subsequences of (an).
    Is the following statement true?
    "Sum of the subsequences is also a subsequence."

    According to me this is wrong. According to the book true.
    When we put n+1 in place of n we get 2n+1
    When we put 2n in place of n we get 4n-1
    If we sum 2n+1+4n-1=6n.So the sum is not a subsequence.
    $a_{2n}=\{-1,3,7,11\cdots\}.$ Starting with $n=0$
    In order to be a subsequence each term must be in the sequence an the order must be preserved.
    Are all of those is the original sequence? Are they in the same order?
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    Re: subsequence

    an=1,3,5.......
    a2n=4n-1=3,7......
    They are in the same order. But as far as I know we can't start with 0.Numbers must be counting numbers not natural numbers.
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    Re: subsequence

    Quote Originally Posted by kastamonu View Post
    an=1,3,5.......
    a2n=4n-1=3,7......
    They are in the same order. But as far as I know we can't start with 0.Numbers must be counting numbers not natural numbers.
    You do not count with zero? What primitive society do you live in?
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    Re: subsequence

    $(a_n)$ is the positive odd numbers

    I'm assuming that $(a_{n+1})_n = a_{n+1}$, this is also a sequence of odd numbers

    Finally $(a_{2n})$ which I assume means $(a_{2n})_n = a_{2n}$ is also a sequence of odd numbers.

    Their sum will be element by element a sum of two odd numbers which is even.

    Thus their sum is a sequence of even numbers.

    Their sum is not a sub-sequence of $(a_n)$
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    Re: subsequence

    Many Thanks for your help.
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    Re: subsequence

    I remember a similar problem from when I took analysis. It said, given two sequences $a_n,b_n $ and any subsequences $(a_i)_{i\in I}, (b_j)_{j\in J}$ where $I,J \subseteq \mathbb{Z}^+$

    Then the sum of the subsequences is a subsequence of the sum of the full sequences. That is as close as I can think to a similar statement that your book would say is true.
    Last edited by SlipEternal; May 20th 2017 at 05:34 AM.
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    Re: subsequence

    It is true but for this problem it is wrong.
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  9. #9
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    Re: subsequence

    Quote Originally Posted by kastamonu View Post
    It is true but for this problem it is wrong.
    That's because I stated the theorem wrong... It has been so many years since I took analysis... I'll try to find the actual statement.
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