Question: Is Ayn Rand 's approach to reasoning purely inductive? If so, is inductive reasoning a philosopical weakness? 

Answer: Induction is the process of reaching general conclusions from particular facts. It is contrasted in logic with deduction, which is the process of reaching less general conclusions from broader general conclusions. Inasmuch as our only direct contact with reality is through our sensory (perceptual) awareness of particular facts, all our knowledge, to be worthy of the name, must have an empirical basis. (Not all empirical conclusions are literally inductive: at the least, the axioms of existence, identity, and causality are based in fact, but they are the basis of induction, not derived from it). When we form new knowledge, we must employ induction and deduction together: induction to prove our general conclusions, and deduction to discover their implications and infer testable hypotheses. Furthermore, we can use deduction to prove claims that are hard to test inductively for practical reasons. 

Inductive reasoning is not a philosophical weakness. In fact, deduction, though often thought of as "certain," is dependent at root on induction for the truth of its claims. This is because:
  • There is no a-priori knowledge. All general conclusions ascribing an identity to a class of things can only be proven relative to the particular things of that class, i.e., by induction.
  • Since the truth of all deductive inferences depends entirely on the truth of their premises, all deductive conclusions depend for their truth on the inductive proof of their most general premises.
So any philosophy that claimed to avoid induction and rely on deduction would be nothing but a house of cards, an idle fantasy for all its logical coherence.
Ayn Rand 's approach to reasoning is not "purely inductive." But induction is essential to it and induction enters her philosophy in every argument. Her full philosophy is a system of inductive generalizations and deductive connections.
You can find a general discussion of the nature of her system in the introduction to the " Logical Structure of Objectivism " beta draft.

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