Finding the common difference and hence, the sum of an A.P












0















Find the sum to $25$ terms of an A.P with the first four terms as $1, log_yx, log_zy,-15log_x z$.




My attempt:
I started out with,



$2log_yx = 1+log_zy$



and,



$2log_zy = log_yx -15log_xz$



Further simplifying the equations led me nowhere. The work was getting too tedious. And since the exam in which I was supposed to solve this question only gives exactly 180 sec to solve this problem, I thought there might be some other, smarter way.



Any help would be appreciated.






sequences-and-series arithmetic-progressions







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    yesterday
















0















Find the sum to $25$ terms of an A.P with the first four terms as $1, log_yx, log_zy,-15log_x z$.




My attempt:
I started out with,



$2log_yx = 1+log_zy$



and,



$2log_zy = log_yx -15log_xz$



Further simplifying the equations led me nowhere. The work was getting too tedious. And since the exam in which I was supposed to solve this question only gives exactly 180 sec to solve this problem, I thought there might be some other, smarter way.



Any help would be appreciated.






sequences-and-series arithmetic-progressions







share|cite|improve this question







New contributor




Sahil Baori is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
















  • 1




    math.stackexchange.com/questions/3060501/…
    – lab bhattacharjee
    yesterday














0












0








0








Find the sum to $25$ terms of an A.P with the first four terms as $1, log_yx, log_zy,-15log_x z$.




My attempt:
I started out with,



$2log_yx = 1+log_zy$



and,



$2log_zy = log_yx -15log_xz$



Further simplifying the equations led me nowhere. The work was getting too tedious. And since the exam in which I was supposed to solve this question only gives exactly 180 sec to solve this problem, I thought there might be some other, smarter way.



Any help would be appreciated.






sequences-and-series arithmetic-progressions







share|cite|improve this question







New contributor




Sahil Baori is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












Find the sum to $25$ terms of an A.P with the first four terms as $1, log_yx, log_zy,-15log_x z$.




My attempt:
I started out with,



$2log_yx = 1+log_zy$



and,



$2log_zy = log_yx -15log_xz$



Further simplifying the equations led me nowhere. The work was getting too tedious. And since the exam in which I was supposed to solve this question only gives exactly 180 sec to solve this problem, I thought there might be some other, smarter way.



Any help would be appreciated.






sequences-and-series arithmetic-progressions




sequences-and-series arithmetic-progressions






share|cite|improve this question







New contributor




Sahil Baori is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question







New contributor




Sahil Baori is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question






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asked yesterday









Sahil Baori

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Sahil Baori is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Sahil Baori is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








  • 1




    math.stackexchange.com/questions/3060501/…
    – lab bhattacharjee
    yesterday














  • 1




    math.stackexchange.com/questions/3060501/…
    – lab bhattacharjee
    yesterday








1




1




math.stackexchange.com/questions/3060501/…
– lab bhattacharjee
yesterday




math.stackexchange.com/questions/3060501/…
– lab bhattacharjee
yesterday










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Hint: we have $$frac{ln(x)}{ln(y)}=1+d$$ and $$frac{ln(y)}{ln(z)}=1+2d$$ and $$frac{ln(z)}{ln(x)}=frac{1+2d}{-15}$$ then we get
$$frac{(1+2d)(1+3d)}{-15}=frac{ln(y)}{ln(x)}=frac{1}{1+d}$$ so we obtain
$$(1+2d)(1+3d)(1+d)=-15$$
Can you solve this equation?






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    -1














    Hint: we have $$frac{ln(x)}{ln(y)}=1+d$$ and $$frac{ln(y)}{ln(z)}=1+2d$$ and $$frac{ln(z)}{ln(x)}=frac{1+2d}{-15}$$ then we get
    $$frac{(1+2d)(1+3d)}{-15}=frac{ln(y)}{ln(x)}=frac{1}{1+d}$$ so we obtain
    $$(1+2d)(1+3d)(1+d)=-15$$
    Can you solve this equation?






    share|cite|improve this answer


























      -1














      Hint: we have $$frac{ln(x)}{ln(y)}=1+d$$ and $$frac{ln(y)}{ln(z)}=1+2d$$ and $$frac{ln(z)}{ln(x)}=frac{1+2d}{-15}$$ then we get
      $$frac{(1+2d)(1+3d)}{-15}=frac{ln(y)}{ln(x)}=frac{1}{1+d}$$ so we obtain
      $$(1+2d)(1+3d)(1+d)=-15$$
      Can you solve this equation?






      share|cite|improve this answer
























        -1












        -1








        -1






        Hint: we have $$frac{ln(x)}{ln(y)}=1+d$$ and $$frac{ln(y)}{ln(z)}=1+2d$$ and $$frac{ln(z)}{ln(x)}=frac{1+2d}{-15}$$ then we get
        $$frac{(1+2d)(1+3d)}{-15}=frac{ln(y)}{ln(x)}=frac{1}{1+d}$$ so we obtain
        $$(1+2d)(1+3d)(1+d)=-15$$
        Can you solve this equation?






        share|cite|improve this answer












        Hint: we have $$frac{ln(x)}{ln(y)}=1+d$$ and $$frac{ln(y)}{ln(z)}=1+2d$$ and $$frac{ln(z)}{ln(x)}=frac{1+2d}{-15}$$ then we get
        $$frac{(1+2d)(1+3d)}{-15}=frac{ln(y)}{ln(x)}=frac{1}{1+d}$$ so we obtain
        $$(1+2d)(1+3d)(1+d)=-15$$
        Can you solve this equation?







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered yesterday









        Dr. Sonnhard Graubner

        73.3k42865




        73.3k42865






















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