Find the smallest possible value of the sum $x_1+x_2+…+x_{2008}$












3












$begingroup$


Let $x_1, x_2,...,x_{2008}$ are numbers such that $|x_1|=999$ and for all $n=2,...,2008$
$$|x_n|=|x_{n-1}+1|$$
Find the smallest possible value of the sum
$$x_1+x_2+...+x_{2008}$$



My work:



Let $S=x_1+x_2+...+x_{2008}$.



If $x_1=-999, x_2=-998, ..., -1,0,-1,0$
then I think the answer $-500004$.



But I don't know how to prove that:






sequences-and-series







share|cite|improve this question











$endgroup$












  • $begingroup$
    Finding the smallest possible value of $|x_1+x_2+cdots+x_{2008}|$ is also an interesting problem. (In fact, it's how I interpreted the problem on first reading.)
    $endgroup$
    – TonyK
    Jan 16 at 12:58












  • $begingroup$
    The minimization problem can be understood as a multi-staging process in which the transition to a new staging, from $sum_{k=1}^N x_k$ to $sum_{k=1}^{N+1} x_k$ is constrained by the condition $ |x_{k+1}| = |x_k+1|$ so the procedure to solve it can be successfully handled with a Dynamic Programming algorithm.
    $endgroup$
    – Cesareo
    Jan 17 at 11:04










  • $begingroup$
    On a side note, @Roman, let me ask you this: aren't you going to create an OEIS entry for the sequence from math.stackexchange.com/questions/2469058/…? IMHO, it more than deserves one.
    $endgroup$
    – Ivan Neretin
    Jan 17 at 22:46










  • $begingroup$
    @IvanNeretin: No! I do not know how do it. But if you want you will can do it.
    $endgroup$
    – Roman83
    Jan 18 at 8:08










  • $begingroup$
    @Roman So I will. May I know your full name, at least? I don't want to claim it in my name, as the idea is yours.
    $endgroup$
    – Ivan Neretin
    Jan 18 at 8:39
















3












$begingroup$


Let $x_1, x_2,...,x_{2008}$ are numbers such that $|x_1|=999$ and for all $n=2,...,2008$
$$|x_n|=|x_{n-1}+1|$$
Find the smallest possible value of the sum
$$x_1+x_2+...+x_{2008}$$



My work:



Let $S=x_1+x_2+...+x_{2008}$.



If $x_1=-999, x_2=-998, ..., -1,0,-1,0$
then I think the answer $-500004$.



But I don't know how to prove that:






sequences-and-series







share|cite|improve this question











$endgroup$












  • $begingroup$
    Finding the smallest possible value of $|x_1+x_2+cdots+x_{2008}|$ is also an interesting problem. (In fact, it's how I interpreted the problem on first reading.)
    $endgroup$
    – TonyK
    Jan 16 at 12:58












  • $begingroup$
    The minimization problem can be understood as a multi-staging process in which the transition to a new staging, from $sum_{k=1}^N x_k$ to $sum_{k=1}^{N+1} x_k$ is constrained by the condition $ |x_{k+1}| = |x_k+1|$ so the procedure to solve it can be successfully handled with a Dynamic Programming algorithm.
    $endgroup$
    – Cesareo
    Jan 17 at 11:04










  • $begingroup$
    On a side note, @Roman, let me ask you this: aren't you going to create an OEIS entry for the sequence from math.stackexchange.com/questions/2469058/…? IMHO, it more than deserves one.
    $endgroup$
    – Ivan Neretin
    Jan 17 at 22:46










  • $begingroup$
    @IvanNeretin: No! I do not know how do it. But if you want you will can do it.
    $endgroup$
    – Roman83
    Jan 18 at 8:08










  • $begingroup$
    @Roman So I will. May I know your full name, at least? I don't want to claim it in my name, as the idea is yours.
    $endgroup$
    – Ivan Neretin
    Jan 18 at 8:39














3












3








3


1



$begingroup$


Let $x_1, x_2,...,x_{2008}$ are numbers such that $|x_1|=999$ and for all $n=2,...,2008$
$$|x_n|=|x_{n-1}+1|$$
Find the smallest possible value of the sum
$$x_1+x_2+...+x_{2008}$$



My work:



Let $S=x_1+x_2+...+x_{2008}$.



If $x_1=-999, x_2=-998, ..., -1,0,-1,0$
then I think the answer $-500004$.



But I don't know how to prove that:






sequences-and-series







share|cite|improve this question











$endgroup$




Let $x_1, x_2,...,x_{2008}$ are numbers such that $|x_1|=999$ and for all $n=2,...,2008$
$$|x_n|=|x_{n-1}+1|$$
Find the smallest possible value of the sum
$$x_1+x_2+...+x_{2008}$$



My work:



Let $S=x_1+x_2+...+x_{2008}$.



If $x_1=-999, x_2=-998, ..., -1,0,-1,0$
then I think the answer $-500004$.



But I don't know how to prove that:






sequences-and-series




sequences-and-series






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 16 at 12:51









user376343

3,9584829




3,9584829










asked Jan 16 at 12:47









Roman83Roman83

14.4k31956




14.4k31956












  • $begingroup$
    Finding the smallest possible value of $|x_1+x_2+cdots+x_{2008}|$ is also an interesting problem. (In fact, it's how I interpreted the problem on first reading.)
    $endgroup$
    – TonyK
    Jan 16 at 12:58












  • $begingroup$
    The minimization problem can be understood as a multi-staging process in which the transition to a new staging, from $sum_{k=1}^N x_k$ to $sum_{k=1}^{N+1} x_k$ is constrained by the condition $ |x_{k+1}| = |x_k+1|$ so the procedure to solve it can be successfully handled with a Dynamic Programming algorithm.
    $endgroup$
    – Cesareo
    Jan 17 at 11:04










  • $begingroup$
    On a side note, @Roman, let me ask you this: aren't you going to create an OEIS entry for the sequence from math.stackexchange.com/questions/2469058/…? IMHO, it more than deserves one.
    $endgroup$
    – Ivan Neretin
    Jan 17 at 22:46










  • $begingroup$
    @IvanNeretin: No! I do not know how do it. But if you want you will can do it.
    $endgroup$
    – Roman83
    Jan 18 at 8:08










  • $begingroup$
    @Roman So I will. May I know your full name, at least? I don't want to claim it in my name, as the idea is yours.
    $endgroup$
    – Ivan Neretin
    Jan 18 at 8:39


















  • $begingroup$
    Finding the smallest possible value of $|x_1+x_2+cdots+x_{2008}|$ is also an interesting problem. (In fact, it's how I interpreted the problem on first reading.)
    $endgroup$
    – TonyK
    Jan 16 at 12:58












  • $begingroup$
    The minimization problem can be understood as a multi-staging process in which the transition to a new staging, from $sum_{k=1}^N x_k$ to $sum_{k=1}^{N+1} x_k$ is constrained by the condition $ |x_{k+1}| = |x_k+1|$ so the procedure to solve it can be successfully handled with a Dynamic Programming algorithm.
    $endgroup$
    – Cesareo
    Jan 17 at 11:04










  • $begingroup$
    On a side note, @Roman, let me ask you this: aren't you going to create an OEIS entry for the sequence from math.stackexchange.com/questions/2469058/…? IMHO, it more than deserves one.
    $endgroup$
    – Ivan Neretin
    Jan 17 at 22:46










  • $begingroup$
    @IvanNeretin: No! I do not know how do it. But if you want you will can do it.
    $endgroup$
    – Roman83
    Jan 18 at 8:08










  • $begingroup$
    @Roman So I will. May I know your full name, at least? I don't want to claim it in my name, as the idea is yours.
    $endgroup$
    – Ivan Neretin
    Jan 18 at 8:39
















$begingroup$
Finding the smallest possible value of $|x_1+x_2+cdots+x_{2008}|$ is also an interesting problem. (In fact, it's how I interpreted the problem on first reading.)
$endgroup$
– TonyK
Jan 16 at 12:58






$begingroup$
Finding the smallest possible value of $|x_1+x_2+cdots+x_{2008}|$ is also an interesting problem. (In fact, it's how I interpreted the problem on first reading.)
$endgroup$
– TonyK
Jan 16 at 12:58














$begingroup$
The minimization problem can be understood as a multi-staging process in which the transition to a new staging, from $sum_{k=1}^N x_k$ to $sum_{k=1}^{N+1} x_k$ is constrained by the condition $ |x_{k+1}| = |x_k+1|$ so the procedure to solve it can be successfully handled with a Dynamic Programming algorithm.
$endgroup$
– Cesareo
Jan 17 at 11:04




$begingroup$
The minimization problem can be understood as a multi-staging process in which the transition to a new staging, from $sum_{k=1}^N x_k$ to $sum_{k=1}^{N+1} x_k$ is constrained by the condition $ |x_{k+1}| = |x_k+1|$ so the procedure to solve it can be successfully handled with a Dynamic Programming algorithm.
$endgroup$
– Cesareo
Jan 17 at 11:04












$begingroup$
On a side note, @Roman, let me ask you this: aren't you going to create an OEIS entry for the sequence from math.stackexchange.com/questions/2469058/…? IMHO, it more than deserves one.
$endgroup$
– Ivan Neretin
Jan 17 at 22:46




$begingroup$
On a side note, @Roman, let me ask you this: aren't you going to create an OEIS entry for the sequence from math.stackexchange.com/questions/2469058/…? IMHO, it more than deserves one.
$endgroup$
– Ivan Neretin
Jan 17 at 22:46












$begingroup$
@IvanNeretin: No! I do not know how do it. But if you want you will can do it.
$endgroup$
– Roman83
Jan 18 at 8:08




$begingroup$
@IvanNeretin: No! I do not know how do it. But if you want you will can do it.
$endgroup$
– Roman83
Jan 18 at 8:08












$begingroup$
@Roman So I will. May I know your full name, at least? I don't want to claim it in my name, as the idea is yours.
$endgroup$
– Ivan Neretin
Jan 18 at 8:39




$begingroup$
@Roman So I will. May I know your full name, at least? I don't want to claim it in my name, as the idea is yours.
$endgroup$
– Ivan Neretin
Jan 18 at 8:39










1 Answer
1






active

oldest

votes


















2












$begingroup$

Hint.



Calling



$$
S_1(x_1) = {-m,m}
$$



and



$$
S_k(x_k) = min_{x_k}left(S_{k-1}(x_{k-1})+x_kright) mbox{s. t. } |x_k|=|x_{k-1}+1|
$$



we have after $N$ steps the sought min as $min S_N(x_N)$






share|cite|improve this answer









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    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    2












    $begingroup$

    Hint.



    Calling



    $$
    S_1(x_1) = {-m,m}
    $$



    and



    $$
    S_k(x_k) = min_{x_k}left(S_{k-1}(x_{k-1})+x_kright) mbox{s. t. } |x_k|=|x_{k-1}+1|
    $$



    we have after $N$ steps the sought min as $min S_N(x_N)$






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      Hint.



      Calling



      $$
      S_1(x_1) = {-m,m}
      $$



      and



      $$
      S_k(x_k) = min_{x_k}left(S_{k-1}(x_{k-1})+x_kright) mbox{s. t. } |x_k|=|x_{k-1}+1|
      $$



      we have after $N$ steps the sought min as $min S_N(x_N)$






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        Hint.



        Calling



        $$
        S_1(x_1) = {-m,m}
        $$



        and



        $$
        S_k(x_k) = min_{x_k}left(S_{k-1}(x_{k-1})+x_kright) mbox{s. t. } |x_k|=|x_{k-1}+1|
        $$



        we have after $N$ steps the sought min as $min S_N(x_N)$






        share|cite|improve this answer









        $endgroup$



        Hint.



        Calling



        $$
        S_1(x_1) = {-m,m}
        $$



        and



        $$
        S_k(x_k) = min_{x_k}left(S_{k-1}(x_{k-1})+x_kright) mbox{s. t. } |x_k|=|x_{k-1}+1|
        $$



        we have after $N$ steps the sought min as $min S_N(x_N)$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 17 at 12:18









        CesareoCesareo

        9,6723517




        9,6723517






























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