Question

1.Determinati n∈N, astfel incat fractia
$\frac{2n+7}{3n+1}$, ,sa fie echiunitara.
2.Determinati n∈N, astfel incat fractia $\frac{1+2+ 2^{2+....+ 2^{100} } }{ 2^{n} }$ sa fie subunitara

Answer 1

    
$\displaystyle 1) \\ \frac{2n+7}{3n+1} =1 \\ \\ 3n+1=2n+7 \\ 3n-2n=7-1 \\ n=\boxed{6} \\ \\ \\ 2) \\ \frac{1+2+2^2+...+2^{100}}{2^n} \ \textless \ 0 \\ \\ 1+2+2^2+...+2^{100}\ \textless \ 2^n \\ 2^{101}-1 \ \textless \ 2^n ~~~~~=\ \textgreater \ \;\;\; \boxed{n \geq 101}$