A recurrence sequence I can't solve (A level) - Any hint is appreciated.












0














A sequence is defined by
$$begin{aligned} u_1 &= 3 \
u_{n+1} &= 2 — 4/u_n
end{aligned}$$



Find the exact values of



(a) $u_2, u_3 ;text{and}; u_4$



(b) $u_{61}$



(c) $sum_{i=1}^{99} u_i$



How to solve (b) and (c) without solving the preceding terms leading up to $u_{60}$ individually ?










share|cite|improve this question






















  • What have you tried so far? Have you done a)?
    – Mindlack
    Jan 6 at 17:51










  • Yes I have, this is the first time I have encountered this type of question, (a) I am familiar with. Feeling really dumb.
    – Antonio
    Jan 6 at 17:52












  • No need to feel dumb, you have to struggle with one problem of that kind and afterwards you’ll master all others! As the answer said, look for patterns. If you do not see it yet, try computing $u_5$, $u_6$, $u_7$, and so on, till you get it (and you will!).
    – Mindlack
    Jan 6 at 17:54
















0














A sequence is defined by
$$begin{aligned} u_1 &= 3 \
u_{n+1} &= 2 — 4/u_n
end{aligned}$$



Find the exact values of



(a) $u_2, u_3 ;text{and}; u_4$



(b) $u_{61}$



(c) $sum_{i=1}^{99} u_i$



How to solve (b) and (c) without solving the preceding terms leading up to $u_{60}$ individually ?










share|cite|improve this question






















  • What have you tried so far? Have you done a)?
    – Mindlack
    Jan 6 at 17:51










  • Yes I have, this is the first time I have encountered this type of question, (a) I am familiar with. Feeling really dumb.
    – Antonio
    Jan 6 at 17:52












  • No need to feel dumb, you have to struggle with one problem of that kind and afterwards you’ll master all others! As the answer said, look for patterns. If you do not see it yet, try computing $u_5$, $u_6$, $u_7$, and so on, till you get it (and you will!).
    – Mindlack
    Jan 6 at 17:54














0












0








0







A sequence is defined by
$$begin{aligned} u_1 &= 3 \
u_{n+1} &= 2 — 4/u_n
end{aligned}$$



Find the exact values of



(a) $u_2, u_3 ;text{and}; u_4$



(b) $u_{61}$



(c) $sum_{i=1}^{99} u_i$



How to solve (b) and (c) without solving the preceding terms leading up to $u_{60}$ individually ?










share|cite|improve this question













A sequence is defined by
$$begin{aligned} u_1 &= 3 \
u_{n+1} &= 2 — 4/u_n
end{aligned}$$



Find the exact values of



(a) $u_2, u_3 ;text{and}; u_4$



(b) $u_{61}$



(c) $sum_{i=1}^{99} u_i$



How to solve (b) and (c) without solving the preceding terms leading up to $u_{60}$ individually ?







sequences-and-series






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 6 at 17:46









AntonioAntonio

11




11












  • What have you tried so far? Have you done a)?
    – Mindlack
    Jan 6 at 17:51










  • Yes I have, this is the first time I have encountered this type of question, (a) I am familiar with. Feeling really dumb.
    – Antonio
    Jan 6 at 17:52












  • No need to feel dumb, you have to struggle with one problem of that kind and afterwards you’ll master all others! As the answer said, look for patterns. If you do not see it yet, try computing $u_5$, $u_6$, $u_7$, and so on, till you get it (and you will!).
    – Mindlack
    Jan 6 at 17:54


















  • What have you tried so far? Have you done a)?
    – Mindlack
    Jan 6 at 17:51










  • Yes I have, this is the first time I have encountered this type of question, (a) I am familiar with. Feeling really dumb.
    – Antonio
    Jan 6 at 17:52












  • No need to feel dumb, you have to struggle with one problem of that kind and afterwards you’ll master all others! As the answer said, look for patterns. If you do not see it yet, try computing $u_5$, $u_6$, $u_7$, and so on, till you get it (and you will!).
    – Mindlack
    Jan 6 at 17:54
















What have you tried so far? Have you done a)?
– Mindlack
Jan 6 at 17:51




What have you tried so far? Have you done a)?
– Mindlack
Jan 6 at 17:51












Yes I have, this is the first time I have encountered this type of question, (a) I am familiar with. Feeling really dumb.
– Antonio
Jan 6 at 17:52






Yes I have, this is the first time I have encountered this type of question, (a) I am familiar with. Feeling really dumb.
– Antonio
Jan 6 at 17:52














No need to feel dumb, you have to struggle with one problem of that kind and afterwards you’ll master all others! As the answer said, look for patterns. If you do not see it yet, try computing $u_5$, $u_6$, $u_7$, and so on, till you get it (and you will!).
– Mindlack
Jan 6 at 17:54




No need to feel dumb, you have to struggle with one problem of that kind and afterwards you’ll master all others! As the answer said, look for patterns. If you do not see it yet, try computing $u_5$, $u_6$, $u_7$, and so on, till you get it (and you will!).
– Mindlack
Jan 6 at 17:54










1 Answer
1






active

oldest

votes


















2














For part (b) you need to look for patterns.
$$u_1=3$$
$$u_2=frac{2}{3}$$
$$u_3=-4$$
$$u_4=3$$
$$u_5=frac{2}{3}$$
$$u_6=-4$$
(and so on)



Do you remember how to find $i^{87}$, where $i=sqrt{-1}$? Same deal.



For part (c) you will need the identity $sum_{k=1}^nc=cn$.






share|cite|improve this answer























    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%2f3064171%2fa-recurrence-sequence-i-cant-solve-a-level-any-hint-is-appreciated%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














    For part (b) you need to look for patterns.
    $$u_1=3$$
    $$u_2=frac{2}{3}$$
    $$u_3=-4$$
    $$u_4=3$$
    $$u_5=frac{2}{3}$$
    $$u_6=-4$$
    (and so on)



    Do you remember how to find $i^{87}$, where $i=sqrt{-1}$? Same deal.



    For part (c) you will need the identity $sum_{k=1}^nc=cn$.






    share|cite|improve this answer




























      2














      For part (b) you need to look for patterns.
      $$u_1=3$$
      $$u_2=frac{2}{3}$$
      $$u_3=-4$$
      $$u_4=3$$
      $$u_5=frac{2}{3}$$
      $$u_6=-4$$
      (and so on)



      Do you remember how to find $i^{87}$, where $i=sqrt{-1}$? Same deal.



      For part (c) you will need the identity $sum_{k=1}^nc=cn$.






      share|cite|improve this answer


























        2












        2








        2






        For part (b) you need to look for patterns.
        $$u_1=3$$
        $$u_2=frac{2}{3}$$
        $$u_3=-4$$
        $$u_4=3$$
        $$u_5=frac{2}{3}$$
        $$u_6=-4$$
        (and so on)



        Do you remember how to find $i^{87}$, where $i=sqrt{-1}$? Same deal.



        For part (c) you will need the identity $sum_{k=1}^nc=cn$.






        share|cite|improve this answer














        For part (b) you need to look for patterns.
        $$u_1=3$$
        $$u_2=frac{2}{3}$$
        $$u_3=-4$$
        $$u_4=3$$
        $$u_5=frac{2}{3}$$
        $$u_6=-4$$
        (and so on)



        Do you remember how to find $i^{87}$, where $i=sqrt{-1}$? Same deal.



        For part (c) you will need the identity $sum_{k=1}^nc=cn$.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Jan 6 at 18:02

























        answered Jan 6 at 17:50









        Ben WBen W

        2,017615




        2,017615






























            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%2f3064171%2fa-recurrence-sequence-i-cant-solve-a-level-any-hint-is-appreciated%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

            Mario Kart Wii

            The Binding of Isaac: Rebirth/Afterbirth

            What does “Dominus providebit” mean?