Question

va rog dau coroana...trebuie să se afle nr. nat. n​

Answer 1

$\displaystyle\bf\\\boxed{\bf o~fractie~este~echiunitara~daca~si~numai~daca~numitorul~este~egal~cu~numaratorul}~.\\\\-------------------------------------\\\\\frac{3^n+3^{n+2}}{2^n+2^{n+1}+3\cdot2^{n+2}}~este~echiunitara~\Leftrightarrow\\3^n+3^{n+2}=2^n+2^{n+1}+3\cdot2^{n+2} \Leftrightarrow\\3^n+9\cdot3^n=2^n+2\cdot2^n+12\cdot2^n \Leftrightarrow\\10\cdot3^n=15\cdot2^n~\bigg |~:5 \implies 2\cdot3^n=3\cdot2^n \Leftrightarrow\\2\cdot \bigg ( \frac{3}{2} \bigg)^n=3 |~:2 \Leftrightarrow$$\displaystyle\bf\\ \bigg ( \frac{3}{2} \bigg)^n=\frac{3}{2},~de~unde~evident~\boxed{\bf n=1}~.$