Question

Cuántos números enteros positivos n satisfacen la siguiente condición:

$(130n)^{50} \ \textgreater \ n^{100} \ \textgreater \ 2^{200}$

Answer 1

(1) si n > 0

$n^{100}\ \textgreater \ 2^{200}\\ \\ n\ \textgreater \ 4$

(2) 

$(130n)^{50} \ \textgreater \ 2^{200} \\ \\ 130n\ \textgreater \ 16\\ \\ n\ \textgreater \ \dfrac{16}{130}$

(3)

$(130n)^{50}\ \textgreater \ n^{100}\\ \\ 130n\ \textgreater \ n^2\\ \\ n\in(0,130)$

(4) (1) & (2) & (3): $n\in\{5,6,7,\cdots,129\}\to \boxed{\#n=125}$