Find all pairs of numbers, which meet the assumption.












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I have to find all pairs of numbers $ c, d in mathbb{R}^2 $, which meet the assumption.



The assumption is:



$ S_n = (c-5d)n^2+ncd^2-2c^2-2d^2+c+d $



is the sum of the first n words of some arithmetic sequence.



So I wrote a formula for the sum of the first n words of arithmetic sequence;



$ S_n= {2a_1n+n^2r-rnover 2} $



Where $n$ is the number of numbers of the arithmetical sequence, $r$ is difference and $a_1$ is the first number.



So now I have:



$2(c-5d)=r $



$2cd^2=2a_1-r$



What should do now? I do not want to insert a number as $r$ or $a_1$ or even $n$.



Thanks in advice










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    I have to find all pairs of numbers $ c, d in mathbb{R}^2 $, which meet the assumption.



    The assumption is:



    $ S_n = (c-5d)n^2+ncd^2-2c^2-2d^2+c+d $



    is the sum of the first n words of some arithmetic sequence.



    So I wrote a formula for the sum of the first n words of arithmetic sequence;



    $ S_n= {2a_1n+n^2r-rnover 2} $



    Where $n$ is the number of numbers of the arithmetical sequence, $r$ is difference and $a_1$ is the first number.



    So now I have:



    $2(c-5d)=r $



    $2cd^2=2a_1-r$



    What should do now? I do not want to insert a number as $r$ or $a_1$ or even $n$.



    Thanks in advice










    share|cite|improve this question



























      0












      0








      0







      I have to find all pairs of numbers $ c, d in mathbb{R}^2 $, which meet the assumption.



      The assumption is:



      $ S_n = (c-5d)n^2+ncd^2-2c^2-2d^2+c+d $



      is the sum of the first n words of some arithmetic sequence.



      So I wrote a formula for the sum of the first n words of arithmetic sequence;



      $ S_n= {2a_1n+n^2r-rnover 2} $



      Where $n$ is the number of numbers of the arithmetical sequence, $r$ is difference and $a_1$ is the first number.



      So now I have:



      $2(c-5d)=r $



      $2cd^2=2a_1-r$



      What should do now? I do not want to insert a number as $r$ or $a_1$ or even $n$.



      Thanks in advice










      share|cite|improve this question















      I have to find all pairs of numbers $ c, d in mathbb{R}^2 $, which meet the assumption.



      The assumption is:



      $ S_n = (c-5d)n^2+ncd^2-2c^2-2d^2+c+d $



      is the sum of the first n words of some arithmetic sequence.



      So I wrote a formula for the sum of the first n words of arithmetic sequence;



      $ S_n= {2a_1n+n^2r-rnover 2} $



      Where $n$ is the number of numbers of the arithmetical sequence, $r$ is difference and $a_1$ is the first number.



      So now I have:



      $2(c-5d)=r $



      $2cd^2=2a_1-r$



      What should do now? I do not want to insert a number as $r$ or $a_1$ or even $n$.



      Thanks in advice







      sequences-and-series systems-of-equations arithmetic square-numbers






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      share|cite|improve this question













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      edited Dec 5 '18 at 22:01

























      asked Dec 2 '18 at 13:12









      Kukoz

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