$\begin{aligned}[t]
u_n>2u_0 & \iff \numprint{75500}\times1,014^n-500>\numprint{150000}\\
& \iff \numprint{75500}\times1,014^n > \numprint{150500}\\
& \iff 1,014^n > \dfrac{\numprint{1505}}{755}\\
& \iff \ln\big(1,014^n\big) > \ln\left(\dfrac{\numprint{1505}}{755}\right)\\
& \iff n\ln(1,014)>\ln\left(\dfrac{\numprint{1505}}{755}\right)\\
& \iff n>\dfrac{\ln\left(\frac{\numprint{1505}}{755}\right)}{\ln(1,014)}\quad\text{ car }\ln(1,014)>0.
\end{aligned}$

\Par

On obtient donc, en arrondissant à l'entier supérieur, $n = 50$.
