Series of Squares Under Triangle

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
1
down vote

favorite












A line crosses the $x$ and $y$ axes at $(a,0)$ and $(0,1)$ respectively, where $a>0$. Square are placed successively inside the right angled triangle thus formed. What is the area enclosed by all squares when their number goes to infinity.



enter image description here






summation geometric-series







share|cite|improve this question



























    up vote
    1
    down vote

    favorite












    A line crosses the $x$ and $y$ axes at $(a,0)$ and $(0,1)$ respectively, where $a>0$. Square are placed successively inside the right angled triangle thus formed. What is the area enclosed by all squares when their number goes to infinity.



    enter image description here






    summation geometric-series







    share|cite|improve this question

























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      A line crosses the $x$ and $y$ axes at $(a,0)$ and $(0,1)$ respectively, where $a>0$. Square are placed successively inside the right angled triangle thus formed. What is the area enclosed by all squares when their number goes to infinity.



      enter image description here






      summation geometric-series







      share|cite|improve this question















      A line crosses the $x$ and $y$ axes at $(a,0)$ and $(0,1)$ respectively, where $a>0$. Square are placed successively inside the right angled triangle thus formed. What is the area enclosed by all squares when their number goes to infinity.



      enter image description here






      summation geometric-series




      summation geometric-series






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 3 hours ago









      mwt

      546113




      546113










      asked 3 hours ago









      Inquirer

      372210




      372210




















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          5
          down vote



          accepted










          Each square of side length $s$ has as "hat" a right triangle with horizontal leg $s$ and vertical leg $frac sa$. Within a single instance of "square + hat", the square occupies a share of $fracs^2s^2+frac12cdot scdot frac sa=frac11+frac12a=frac2a2a+1$. This proportion remains constant, no matter how many squares with hat we consider, hence also in the limit.
          We conclude that the total area occupied by the squares is
          $$ frac2a2a+1cdot frac a2=fraca^22a+1.$$






          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',
            convertImagesToLinks: true,
            noModals: false,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            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%2f2927780%2fseries-of-squares-under-triangle%23new-answer', 'question_page');

            );

            Post as a guest

















            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            5
            down vote



            accepted










            Each square of side length $s$ has as "hat" a right triangle with horizontal leg $s$ and vertical leg $frac sa$. Within a single instance of "square + hat", the square occupies a share of $fracs^2s^2+frac12cdot scdot frac sa=frac11+frac12a=frac2a2a+1$. This proportion remains constant, no matter how many squares with hat we consider, hence also in the limit.
            We conclude that the total area occupied by the squares is
            $$ frac2a2a+1cdot frac a2=fraca^22a+1.$$






            share|cite|improve this answer
























              up vote
              5
              down vote



              accepted










              Each square of side length $s$ has as "hat" a right triangle with horizontal leg $s$ and vertical leg $frac sa$. Within a single instance of "square + hat", the square occupies a share of $fracs^2s^2+frac12cdot scdot frac sa=frac11+frac12a=frac2a2a+1$. This proportion remains constant, no matter how many squares with hat we consider, hence also in the limit.
              We conclude that the total area occupied by the squares is
              $$ frac2a2a+1cdot frac a2=fraca^22a+1.$$






              share|cite|improve this answer






















                up vote
                5
                down vote



                accepted







                up vote
                5
                down vote



                accepted






                Each square of side length $s$ has as "hat" a right triangle with horizontal leg $s$ and vertical leg $frac sa$. Within a single instance of "square + hat", the square occupies a share of $fracs^2s^2+frac12cdot scdot frac sa=frac11+frac12a=frac2a2a+1$. This proportion remains constant, no matter how many squares with hat we consider, hence also in the limit.
                We conclude that the total area occupied by the squares is
                $$ frac2a2a+1cdot frac a2=fraca^22a+1.$$






                share|cite|improve this answer












                Each square of side length $s$ has as "hat" a right triangle with horizontal leg $s$ and vertical leg $frac sa$. Within a single instance of "square + hat", the square occupies a share of $fracs^2s^2+frac12cdot scdot frac sa=frac11+frac12a=frac2a2a+1$. This proportion remains constant, no matter how many squares with hat we consider, hence also in the limit.
                We conclude that the total area occupied by the squares is
                $$ frac2a2a+1cdot frac a2=fraca^22a+1.$$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 3 hours ago









                Hagen von Eitzen

                268k21260484




                268k21260484



























                     

                    draft saved


                    draft discarded















































                     


                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function ()
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2927780%2fseries-of-squares-under-triangle%23new-answer', 'question_page');

                    );

                    Post as a guest
































































                    Comments

                    Post a Comment

                    Popular posts from this blog

                    Long meetings (6-7 hours a day): Being “babysat” by supervisor

                    Image
                    Clash Royale CLAN TAG #URR8PPP .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty margin-bottom:0; up vote 31 down vote favorite 4 I hold a PhD in mechanical engineering with a 7 year background in self-guided, self-paced research including 2 years as a graduate level course instructor and a subject matter expert for various engineering projects in a university setting in the United States. For the last 10-11 months, I am working in a research industry in France where my task is to debug C++ spaghetti code written over a period of several years by my current supervisor. It is a team of 2: just I and my supervisor. Diurnally, I am posed with a "bug of the day" to get out of this C++ code. My supervisor generally gives me about 15-30 minutes to solve it and then accosts me about whether or not the task is completed and then rinses and repeats this until the end of the day. Off-late, my supervisor has been holding 6-7 hour long "meetin
                    Read more

                    What does second last employer means? [closed]

                    Image
                    Clash Royale CLAN TAG #URR8PPP .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty margin-bottom:0; up vote -3 down vote favorite Say, for example I have worked in the following companies, Company A => 2010 - 2011 Company B => 2011 - 2012 Company C => 2012 - Till Date Company D => Have offer Now assume Company D is asking to submit documents from my second last employer. So from the list of companies given, which company would be qualified as second last employer? software-industry employer share | improve this question asked Aug 10 '14 at 5:16 Rajaprabhu Aravindasamy 559 1 6 13 closed as off-topic by Jim G., jmort253 ♦ Aug 11 '14 at 1:58 This question appears to be off-topic. The users who voted to close gave this specific reason: "Questions seeking advice on company-specific regulations, agreements, or policies should be directed to your manager or HR department. Questions that address only
                    Read more

                    One-line joke

                    A one-liner is a joke that is delivered in a single line. A good one-liner is said to be pithy - concise and meaningful. [1] Comedians and actors use this comedic method as part of their act, e.g. Jimmy Carr, Tommy Cooper, Rodney Dangerfield, Norm Macdonald, Ken Dodd, Stewart Francis, Zach Galifianakis, Mitch Hedberg, Anthony Jeselnik, Milton Jones, Shappi Khorsandi, Jay London, Mark Linn-Baker, Demetri Martin, Groucho Marx, Dan Mintz, Emo Philips, Tim Vine, Steven Wright and Henny Youngman and Arnold Schwarzenegger. [ citation needed ] Many fictional characters are also known to deliver one-liners, including James Bond, who usually includes pithy and laconic quips after disposing of a villain. [ citation needed ] Examples "A baby seal walks into a club." "A dyslexic man walks into a bra." (Anonymous) "There are three types of people, those who can count and those who can't." "Venison's dear, isn't it?" (Jimmy Carr) "An escala
                    Read more