Is it reasonable to use induction to conclude that the universe probably had a beginning?
To use inductive reasonning you need a record of experiences with a similar object than the one you are considering now.
Let's say I have a whole egg in my hand. I know that if I drop it it will break on impacting the floor.
Obviously, I never dropped this particular egg, otherwise it would be broken. But I have dropped lots of similar eggs in the past (I'm super clumsy), and based on this experience I expect this egg to break too. The fact that I have a large record of experiences and that any chicken egg is similar enough to any other makes it reasonable to draw this conclusion.
Based on this, your question can be reformulated as "do I have enough experience of having observed the beginning of objects similar enough to the universe ?". My personal opinion is that no object can be considered similar to the universe, therefore no.