$u_{n+1} = a u_n +b u_{n-1} +c$












2












$begingroup$


Is there a closed form formula for the general term of a recurrence sequence satisfying $$u_{n+1} = a u_n +b u_{n-1} +c$$ The sequence I am interested in satisfies $u_{n+1} = 8 u_n - 3 u_{n-1} -4$.










share|cite|improve this question











$endgroup$

















    2












    $begingroup$


    Is there a closed form formula for the general term of a recurrence sequence satisfying $$u_{n+1} = a u_n +b u_{n-1} +c$$ The sequence I am interested in satisfies $u_{n+1} = 8 u_n - 3 u_{n-1} -4$.










    share|cite|improve this question











    $endgroup$















      2












      2








      2





      $begingroup$


      Is there a closed form formula for the general term of a recurrence sequence satisfying $$u_{n+1} = a u_n +b u_{n-1} +c$$ The sequence I am interested in satisfies $u_{n+1} = 8 u_n - 3 u_{n-1} -4$.










      share|cite|improve this question











      $endgroup$




      Is there a closed form formula for the general term of a recurrence sequence satisfying $$u_{n+1} = a u_n +b u_{n-1} +c$$ The sequence I am interested in satisfies $u_{n+1} = 8 u_n - 3 u_{n-1} -4$.







      real-analysis sequences-and-series recurrence-relations






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 20 '18 at 15:04









      Ankit Kumar

      1,494221




      1,494221










      asked Dec 20 '18 at 13:59









      furyofuryo

      313




      313






















          2 Answers
          2






          active

          oldest

          votes


















          2












          $begingroup$

          You can solve any non-homogeneous, linear recurrence of the above form by converting it to homogeneous form.



          (I'll use $n$ instead of $n+1$, doesn't matter anyway!)



          $$u_{n}=au_{n-1}+bu_{n-2}+c$$
          $$u_{n-1}=au_{n-2}+bu_{n-3}+c$$
          Subtract them to get:-



          $$u_{n}=(a+1)u_{n-1}+(b-a)u_{n-2}-bu_{n-3}$$
          You can then use characteristic polynomial to find the answer.





          Speaking strictly for the equation you wrote, the characteristic polynomial is
          $$z^3=9z^2-11z+3$$ which has the roots
          $$z=1, 4+sqrt{13}, 4-sqrt{13}$$
          giving $$u_n=a_1+a_2{(4+sqrt{13})}^n+a_3{(4-sqrt{13})}^n$$, where $a_i$s depend on the initial condition.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            So $u_n$ wouldn't converge? I find it a little bit strange.
            $endgroup$
            – furyo
            Dec 20 '18 at 16:39










          • $begingroup$
            @furyo It isn't strange at all. Stuff like this (convergence) depends on the initial conditions. Depending upon them, the series may or may not convegre!
            $endgroup$
            – Ankit Kumar
            Dec 20 '18 at 16:43










          • $begingroup$
            For instance, if you take $u_1=u_2=1$, $u_i=1$ for all $i$, which is a convergent sequence. On the other hand, if you take $u_1=1, u_2=2$, then it'll be a divergent sequence.
            $endgroup$
            – Ankit Kumar
            Dec 20 '18 at 16:44










          • $begingroup$
            I would've expected a non constant sequence increasing to $1$. Here it's either constant or divergent.
            $endgroup$
            – furyo
            Dec 20 '18 at 16:47










          • $begingroup$
            You can't get a non-constant, convergent sequence because it isn't constant $implies $ at least one of $a_2, a_3$ is non-zero $implies $ $u_n$ will be a strictly increasing/decreasing (depending upon sign of $a_i$s) function of $n$.
            $endgroup$
            – Ankit Kumar
            Dec 20 '18 at 16:54





















          0












          $begingroup$

          This is a linear difference equation so can be stated as



          $$
          u = u_h + u_p
          $$



          with



          $$
          u_h(n+1)-au_h(n)-bu_h(n-1) = 0\
          u_p(n+1)-au_p(n)-bu_p(n-1) = c\
          $$



          now making $u_h(n) = phi^n$ and substituting we have



          $$
          phi^nleft(phi-a-bphi^{-1}right) = 0
          $$



          so



          $$
          phi = frac{1}{2} left(apmsqrt{a^2+4 b}right)
          $$



          and



          $$
          u_h(n) = frac{C_1}{2^n} left(a-sqrt{a^2+4 b}right)^n+frac{C_2}{2^n} left(a+sqrt{a^2+4 b}right)^n
          $$



          and also



          $$
          u_p(n) = frac{c}{1-a-b}
          $$



          so finally



          $$
          u(n) = frac{C_1}{2^n} left(a-sqrt{a^2+4 b}right)^n+frac{C_2}{2^n} left(a+sqrt{a^2+4 b}right)^n + frac{c}{1-a-b}
          $$






          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%2f3047566%2fu-n1-a-u-n-b-u-n-1-c%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            2












            $begingroup$

            You can solve any non-homogeneous, linear recurrence of the above form by converting it to homogeneous form.



            (I'll use $n$ instead of $n+1$, doesn't matter anyway!)



            $$u_{n}=au_{n-1}+bu_{n-2}+c$$
            $$u_{n-1}=au_{n-2}+bu_{n-3}+c$$
            Subtract them to get:-



            $$u_{n}=(a+1)u_{n-1}+(b-a)u_{n-2}-bu_{n-3}$$
            You can then use characteristic polynomial to find the answer.





            Speaking strictly for the equation you wrote, the characteristic polynomial is
            $$z^3=9z^2-11z+3$$ which has the roots
            $$z=1, 4+sqrt{13}, 4-sqrt{13}$$
            giving $$u_n=a_1+a_2{(4+sqrt{13})}^n+a_3{(4-sqrt{13})}^n$$, where $a_i$s depend on the initial condition.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              So $u_n$ wouldn't converge? I find it a little bit strange.
              $endgroup$
              – furyo
              Dec 20 '18 at 16:39










            • $begingroup$
              @furyo It isn't strange at all. Stuff like this (convergence) depends on the initial conditions. Depending upon them, the series may or may not convegre!
              $endgroup$
              – Ankit Kumar
              Dec 20 '18 at 16:43










            • $begingroup$
              For instance, if you take $u_1=u_2=1$, $u_i=1$ for all $i$, which is a convergent sequence. On the other hand, if you take $u_1=1, u_2=2$, then it'll be a divergent sequence.
              $endgroup$
              – Ankit Kumar
              Dec 20 '18 at 16:44










            • $begingroup$
              I would've expected a non constant sequence increasing to $1$. Here it's either constant or divergent.
              $endgroup$
              – furyo
              Dec 20 '18 at 16:47










            • $begingroup$
              You can't get a non-constant, convergent sequence because it isn't constant $implies $ at least one of $a_2, a_3$ is non-zero $implies $ $u_n$ will be a strictly increasing/decreasing (depending upon sign of $a_i$s) function of $n$.
              $endgroup$
              – Ankit Kumar
              Dec 20 '18 at 16:54


















            2












            $begingroup$

            You can solve any non-homogeneous, linear recurrence of the above form by converting it to homogeneous form.



            (I'll use $n$ instead of $n+1$, doesn't matter anyway!)



            $$u_{n}=au_{n-1}+bu_{n-2}+c$$
            $$u_{n-1}=au_{n-2}+bu_{n-3}+c$$
            Subtract them to get:-



            $$u_{n}=(a+1)u_{n-1}+(b-a)u_{n-2}-bu_{n-3}$$
            You can then use characteristic polynomial to find the answer.





            Speaking strictly for the equation you wrote, the characteristic polynomial is
            $$z^3=9z^2-11z+3$$ which has the roots
            $$z=1, 4+sqrt{13}, 4-sqrt{13}$$
            giving $$u_n=a_1+a_2{(4+sqrt{13})}^n+a_3{(4-sqrt{13})}^n$$, where $a_i$s depend on the initial condition.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              So $u_n$ wouldn't converge? I find it a little bit strange.
              $endgroup$
              – furyo
              Dec 20 '18 at 16:39










            • $begingroup$
              @furyo It isn't strange at all. Stuff like this (convergence) depends on the initial conditions. Depending upon them, the series may or may not convegre!
              $endgroup$
              – Ankit Kumar
              Dec 20 '18 at 16:43










            • $begingroup$
              For instance, if you take $u_1=u_2=1$, $u_i=1$ for all $i$, which is a convergent sequence. On the other hand, if you take $u_1=1, u_2=2$, then it'll be a divergent sequence.
              $endgroup$
              – Ankit Kumar
              Dec 20 '18 at 16:44










            • $begingroup$
              I would've expected a non constant sequence increasing to $1$. Here it's either constant or divergent.
              $endgroup$
              – furyo
              Dec 20 '18 at 16:47










            • $begingroup$
              You can't get a non-constant, convergent sequence because it isn't constant $implies $ at least one of $a_2, a_3$ is non-zero $implies $ $u_n$ will be a strictly increasing/decreasing (depending upon sign of $a_i$s) function of $n$.
              $endgroup$
              – Ankit Kumar
              Dec 20 '18 at 16:54
















            2












            2








            2





            $begingroup$

            You can solve any non-homogeneous, linear recurrence of the above form by converting it to homogeneous form.



            (I'll use $n$ instead of $n+1$, doesn't matter anyway!)



            $$u_{n}=au_{n-1}+bu_{n-2}+c$$
            $$u_{n-1}=au_{n-2}+bu_{n-3}+c$$
            Subtract them to get:-



            $$u_{n}=(a+1)u_{n-1}+(b-a)u_{n-2}-bu_{n-3}$$
            You can then use characteristic polynomial to find the answer.





            Speaking strictly for the equation you wrote, the characteristic polynomial is
            $$z^3=9z^2-11z+3$$ which has the roots
            $$z=1, 4+sqrt{13}, 4-sqrt{13}$$
            giving $$u_n=a_1+a_2{(4+sqrt{13})}^n+a_3{(4-sqrt{13})}^n$$, where $a_i$s depend on the initial condition.






            share|cite|improve this answer









            $endgroup$



            You can solve any non-homogeneous, linear recurrence of the above form by converting it to homogeneous form.



            (I'll use $n$ instead of $n+1$, doesn't matter anyway!)



            $$u_{n}=au_{n-1}+bu_{n-2}+c$$
            $$u_{n-1}=au_{n-2}+bu_{n-3}+c$$
            Subtract them to get:-



            $$u_{n}=(a+1)u_{n-1}+(b-a)u_{n-2}-bu_{n-3}$$
            You can then use characteristic polynomial to find the answer.





            Speaking strictly for the equation you wrote, the characteristic polynomial is
            $$z^3=9z^2-11z+3$$ which has the roots
            $$z=1, 4+sqrt{13}, 4-sqrt{13}$$
            giving $$u_n=a_1+a_2{(4+sqrt{13})}^n+a_3{(4-sqrt{13})}^n$$, where $a_i$s depend on the initial condition.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Dec 20 '18 at 14:24









            Ankit KumarAnkit Kumar

            1,494221




            1,494221












            • $begingroup$
              So $u_n$ wouldn't converge? I find it a little bit strange.
              $endgroup$
              – furyo
              Dec 20 '18 at 16:39










            • $begingroup$
              @furyo It isn't strange at all. Stuff like this (convergence) depends on the initial conditions. Depending upon them, the series may or may not convegre!
              $endgroup$
              – Ankit Kumar
              Dec 20 '18 at 16:43










            • $begingroup$
              For instance, if you take $u_1=u_2=1$, $u_i=1$ for all $i$, which is a convergent sequence. On the other hand, if you take $u_1=1, u_2=2$, then it'll be a divergent sequence.
              $endgroup$
              – Ankit Kumar
              Dec 20 '18 at 16:44










            • $begingroup$
              I would've expected a non constant sequence increasing to $1$. Here it's either constant or divergent.
              $endgroup$
              – furyo
              Dec 20 '18 at 16:47










            • $begingroup$
              You can't get a non-constant, convergent sequence because it isn't constant $implies $ at least one of $a_2, a_3$ is non-zero $implies $ $u_n$ will be a strictly increasing/decreasing (depending upon sign of $a_i$s) function of $n$.
              $endgroup$
              – Ankit Kumar
              Dec 20 '18 at 16:54




















            • $begingroup$
              So $u_n$ wouldn't converge? I find it a little bit strange.
              $endgroup$
              – furyo
              Dec 20 '18 at 16:39










            • $begingroup$
              @furyo It isn't strange at all. Stuff like this (convergence) depends on the initial conditions. Depending upon them, the series may or may not convegre!
              $endgroup$
              – Ankit Kumar
              Dec 20 '18 at 16:43










            • $begingroup$
              For instance, if you take $u_1=u_2=1$, $u_i=1$ for all $i$, which is a convergent sequence. On the other hand, if you take $u_1=1, u_2=2$, then it'll be a divergent sequence.
              $endgroup$
              – Ankit Kumar
              Dec 20 '18 at 16:44










            • $begingroup$
              I would've expected a non constant sequence increasing to $1$. Here it's either constant or divergent.
              $endgroup$
              – furyo
              Dec 20 '18 at 16:47










            • $begingroup$
              You can't get a non-constant, convergent sequence because it isn't constant $implies $ at least one of $a_2, a_3$ is non-zero $implies $ $u_n$ will be a strictly increasing/decreasing (depending upon sign of $a_i$s) function of $n$.
              $endgroup$
              – Ankit Kumar
              Dec 20 '18 at 16:54


















            $begingroup$
            So $u_n$ wouldn't converge? I find it a little bit strange.
            $endgroup$
            – furyo
            Dec 20 '18 at 16:39




            $begingroup$
            So $u_n$ wouldn't converge? I find it a little bit strange.
            $endgroup$
            – furyo
            Dec 20 '18 at 16:39












            $begingroup$
            @furyo It isn't strange at all. Stuff like this (convergence) depends on the initial conditions. Depending upon them, the series may or may not convegre!
            $endgroup$
            – Ankit Kumar
            Dec 20 '18 at 16:43




            $begingroup$
            @furyo It isn't strange at all. Stuff like this (convergence) depends on the initial conditions. Depending upon them, the series may or may not convegre!
            $endgroup$
            – Ankit Kumar
            Dec 20 '18 at 16:43












            $begingroup$
            For instance, if you take $u_1=u_2=1$, $u_i=1$ for all $i$, which is a convergent sequence. On the other hand, if you take $u_1=1, u_2=2$, then it'll be a divergent sequence.
            $endgroup$
            – Ankit Kumar
            Dec 20 '18 at 16:44




            $begingroup$
            For instance, if you take $u_1=u_2=1$, $u_i=1$ for all $i$, which is a convergent sequence. On the other hand, if you take $u_1=1, u_2=2$, then it'll be a divergent sequence.
            $endgroup$
            – Ankit Kumar
            Dec 20 '18 at 16:44












            $begingroup$
            I would've expected a non constant sequence increasing to $1$. Here it's either constant or divergent.
            $endgroup$
            – furyo
            Dec 20 '18 at 16:47




            $begingroup$
            I would've expected a non constant sequence increasing to $1$. Here it's either constant or divergent.
            $endgroup$
            – furyo
            Dec 20 '18 at 16:47












            $begingroup$
            You can't get a non-constant, convergent sequence because it isn't constant $implies $ at least one of $a_2, a_3$ is non-zero $implies $ $u_n$ will be a strictly increasing/decreasing (depending upon sign of $a_i$s) function of $n$.
            $endgroup$
            – Ankit Kumar
            Dec 20 '18 at 16:54






            $begingroup$
            You can't get a non-constant, convergent sequence because it isn't constant $implies $ at least one of $a_2, a_3$ is non-zero $implies $ $u_n$ will be a strictly increasing/decreasing (depending upon sign of $a_i$s) function of $n$.
            $endgroup$
            – Ankit Kumar
            Dec 20 '18 at 16:54













            0












            $begingroup$

            This is a linear difference equation so can be stated as



            $$
            u = u_h + u_p
            $$



            with



            $$
            u_h(n+1)-au_h(n)-bu_h(n-1) = 0\
            u_p(n+1)-au_p(n)-bu_p(n-1) = c\
            $$



            now making $u_h(n) = phi^n$ and substituting we have



            $$
            phi^nleft(phi-a-bphi^{-1}right) = 0
            $$



            so



            $$
            phi = frac{1}{2} left(apmsqrt{a^2+4 b}right)
            $$



            and



            $$
            u_h(n) = frac{C_1}{2^n} left(a-sqrt{a^2+4 b}right)^n+frac{C_2}{2^n} left(a+sqrt{a^2+4 b}right)^n
            $$



            and also



            $$
            u_p(n) = frac{c}{1-a-b}
            $$



            so finally



            $$
            u(n) = frac{C_1}{2^n} left(a-sqrt{a^2+4 b}right)^n+frac{C_2}{2^n} left(a+sqrt{a^2+4 b}right)^n + frac{c}{1-a-b}
            $$






            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$

              This is a linear difference equation so can be stated as



              $$
              u = u_h + u_p
              $$



              with



              $$
              u_h(n+1)-au_h(n)-bu_h(n-1) = 0\
              u_p(n+1)-au_p(n)-bu_p(n-1) = c\
              $$



              now making $u_h(n) = phi^n$ and substituting we have



              $$
              phi^nleft(phi-a-bphi^{-1}right) = 0
              $$



              so



              $$
              phi = frac{1}{2} left(apmsqrt{a^2+4 b}right)
              $$



              and



              $$
              u_h(n) = frac{C_1}{2^n} left(a-sqrt{a^2+4 b}right)^n+frac{C_2}{2^n} left(a+sqrt{a^2+4 b}right)^n
              $$



              and also



              $$
              u_p(n) = frac{c}{1-a-b}
              $$



              so finally



              $$
              u(n) = frac{C_1}{2^n} left(a-sqrt{a^2+4 b}right)^n+frac{C_2}{2^n} left(a+sqrt{a^2+4 b}right)^n + frac{c}{1-a-b}
              $$






              share|cite|improve this answer









              $endgroup$
















                0












                0








                0





                $begingroup$

                This is a linear difference equation so can be stated as



                $$
                u = u_h + u_p
                $$



                with



                $$
                u_h(n+1)-au_h(n)-bu_h(n-1) = 0\
                u_p(n+1)-au_p(n)-bu_p(n-1) = c\
                $$



                now making $u_h(n) = phi^n$ and substituting we have



                $$
                phi^nleft(phi-a-bphi^{-1}right) = 0
                $$



                so



                $$
                phi = frac{1}{2} left(apmsqrt{a^2+4 b}right)
                $$



                and



                $$
                u_h(n) = frac{C_1}{2^n} left(a-sqrt{a^2+4 b}right)^n+frac{C_2}{2^n} left(a+sqrt{a^2+4 b}right)^n
                $$



                and also



                $$
                u_p(n) = frac{c}{1-a-b}
                $$



                so finally



                $$
                u(n) = frac{C_1}{2^n} left(a-sqrt{a^2+4 b}right)^n+frac{C_2}{2^n} left(a+sqrt{a^2+4 b}right)^n + frac{c}{1-a-b}
                $$






                share|cite|improve this answer









                $endgroup$



                This is a linear difference equation so can be stated as



                $$
                u = u_h + u_p
                $$



                with



                $$
                u_h(n+1)-au_h(n)-bu_h(n-1) = 0\
                u_p(n+1)-au_p(n)-bu_p(n-1) = c\
                $$



                now making $u_h(n) = phi^n$ and substituting we have



                $$
                phi^nleft(phi-a-bphi^{-1}right) = 0
                $$



                so



                $$
                phi = frac{1}{2} left(apmsqrt{a^2+4 b}right)
                $$



                and



                $$
                u_h(n) = frac{C_1}{2^n} left(a-sqrt{a^2+4 b}right)^n+frac{C_2}{2^n} left(a+sqrt{a^2+4 b}right)^n
                $$



                and also



                $$
                u_p(n) = frac{c}{1-a-b}
                $$



                so finally



                $$
                u(n) = frac{C_1}{2^n} left(a-sqrt{a^2+4 b}right)^n+frac{C_2}{2^n} left(a+sqrt{a^2+4 b}right)^n + frac{c}{1-a-b}
                $$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 20 '18 at 14:25









                CesareoCesareo

                8,8993516




                8,8993516






























                    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%2f3047566%2fu-n1-a-u-n-b-u-n-1-c%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

                    Ellipse (mathématiques)

                    Quarter-circle Tiles

                    Mont Emei