  We study diffusion-limited pair annihilation $A+A\to 0$ on one-dimensional
lattices with inhomogeneous nearest neighbour hopping in the limit of infinite
reaction rate. We obtain a simple exact expression for the particle
concentration $\rho_k(t)$ of the many-particle system in terms of the
conditional probabilities $P(m;t|l;0)$ for a single random walker in a dual
medium. For some disordered systems with an initially randomly filled lattice
this leads asymptotically to $\bar{\rho(t)}=\bar{P(0;2t|0;0)}$ for the
disorder-averaged particle density. We also obtain interesting exact relations
for single-particle conditional probabilities in random media related by
duality, such as random-barrier and random-trap systems. For some specific
random barrier systems the Smoluchovsky approach to diffusion-limited
annihilation turns out to fail.
