(* k=1 *)
Solve[{(pn+pa)A+pn B1==R,pn A+(pn+pb) B1==R},{A,B1}]//Simplify

(* k=2 *)
Solve[{(pn+pa)A+pn B1+pn B2==R,pn A+(pn+pb) B1+pn B2==R, pn A + pn B1 + (pn+pb) B2==R},{A,B1,B2}]//Simplify

(* k=3 *)
Solve[{(pn+pa)A+pn B1+pn B2+pn B3==R,pn A+(pn+pb) B1+pn B2+pn B3==R, pn A + pn B1 + (pn+pb) B2+pn B3==R, pn A + pn B1 + pn B2+(pn+pb)B3==R},{A,B1,B2,B3}]//Simplify