Solve logistic growth and sketch its changing curvature

A fish population satisfies $dP/dt=0.02P(100-P)$, with $t$ in weeks.

(i) Find the range of $P$ for strictly increasing population. Initially $P=20$.

(ii) Show that $P=100/(1+4e^{-2t})$.

(iii) Describe the long-run population and sketch $P$ against $t$.