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









$endgroup$














    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3075688%2ffind-the-smallest-possible-value-of-the-sum-x-1x-2-x-2008%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    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






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3075688%2ffind-the-smallest-possible-value-of-the-sum-x-1x-2-x-2008%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Human spaceflight

            Image
            Human spaceflight From Wikipedia, the free encyclopedia Jump to navigation Jump to search "Space traveler" redirects here. For other uses, see Space traveler (disambiguation). @media all and (max-width:720px){.mw-parser-output .tmulti>.thumbinner{width:100%!important;max-width:none!important}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:none!important;width:100%!important;text-align:center}} Apollo 11 crewmember Buzz Aldrin on the Moon, 1969 International Space Station crewmember Tracy Caldwell Dyson views the Earth, 2010 Space Shuttle Discovery heads into space with a crew aboard, STS-121 in 2006 Inside a space suit on the Canadarm, 1993 Human spaceflight (also referred to as crewed spaceflight or manned spaceflight ) is space travel with a crew or passengers aboard the spacecraft. Spacecraft carrying people may be operated directly, by human crew, or it may be e
            Read more

            Can not write log (Is /dev/pts mounted?) - openpty in Ubuntu-on-Windows?

            Image
            0 1 What exactly does the error message beginning with E: mean? I assume it has to do with the file structure on Ubuntu-on-Windows, but what exactly? The following package was automatically installed and is no longer required: os-prober Use 'apt-get autoremove' to remove it. The following extra packages will be installed: libxslt1.1 The following NEW packages will be installed: libxslt1.1 xmlstarlet 0 upgraded, 2 newly installed, 0 to remove and 0 not upgraded. Need to get 435 kB of archives. After this operation, 1023 kB of additional disk space will be used. Get:1 http://archive.ubuntu.com/ubuntu/ trusty/main libxslt1.1 amd64 1.1.28-2build1 [145 kB] Get:2 http://archive.ubuntu.com/ubuntu/ trusty/universe xmlstarlet amd64 1.5.0-1 [290 kB]
            Read more

            File:DeusFollowingSea.jpg

            Image
            File:DeusFollowingSea.jpg From Wikipedia, the free encyclopedia Jump to navigation Jump to search File File history File usage No higher resolution available. DeusFollowingSea.jpg ‎ (238 × 211 pixels, file size: 12 KB, MIME type: image/jpeg ) Summary [ edit ] Media data and Non-free use rationale Description Cover art of the Deus CD Following Sea Author or copyright owner Deus Source (WP:NFCC#4) rocksucker.co.uk Date of publication 2012 Use in article (WP:NFCC#7) Following Sea Purpose of use in article (WP:NFCC#8) to serve as the primary means of visual identification at the top of the article dedicated to the work in question. Not replaceable with free media because (WP:NFCC#1) n.a. Minimal use (WP:NFCC#3) Low res image used in infobox of the article on the album Respect for commercial opportunities (WP:NFCC#2) n.a. Fair use Fair use of co
            Read more