[tex]\displaystyle\bf\\\text{folosim formuele:}\\\\1)~~~\log_ab=\frac{1}{\log_ba}\\\\2)~~~\log_ab^c=c\cdot\log_ab\\\\Rezolvare:\\\\\frac{1}{\log_23}+\frac{1}{\log_43}+\cdots+\frac{1}{\log_{2^n}3}=55\log_32\\\\\\log_32+log_34+\cdots+log_32^n=55log_32\\\\log_32^1+log_32^2+\cdots+log_32^n=55log_32\\\\1\times\log_32+2\times\log_32+\cdots+n\log_32=55log_32[/tex]
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[tex]\displaystyle\bf\\Dam~factor~comun.\\\\(1+2+\cdots~+n)log_32=55log_32~~~~~\Big|:log_32\\\\1+2+\cdots~+n=55\\\\\\\frac{n(n+1)}{2}=55\\\\\\n(n+1)=55\times2\\\\n(n+1)=110\\\\n(n+1)~~este~produs~de~numere~consecutive.\\\\\text{\bf Descompunem numarul 110 in produs de numere consecutive.}\\\\110=10\times11\\\\n(n+1)=10\times11\\\\\implies~\boxed{\bf~n=10}\\\\Verificare:\\\\1+2+3+\cdots+10=\frac{10(10+1)}{2}=5\times11=55~~corect.[/tex]
