In the generalized random-walk problem of section 8 put [in analogy with (8.1)] P Pa+Pa+1-2 +Pa+2-2 +, and let dn be the probability that the game lasts for exactly n steps. Show that for n 1 a-1 dz,n+1 = dx,nPx-z with dz.1 rpz. Hence prove that the generating function d(o) = dz,non is the solution of the system of linear equations 'd; (0) - d(o)p = rx + Pz- x=1 By differentiation it follows that the expected duration ez is the solution of - 2-1 exPx-x = 1.