Which was why Hume, in the end, argued that since the question of justification was futile (since there could be no justification that did not beg the question) that our best bet was to become natural epistemologists and inquire into the the psychological nature of our use of induction.
And having inquired into the psychological nature of our use of induction, what should our conclusion be? Is it rational to continue using induction while believing that it has no adequate justification?
"Rational" may simply mean that we find inductive inference useful so far, even if we have no good reason to think that the next time it will work. Especially since there is nothing better. If I need something to bang a nail into the wood, and if I don't have a hammer, but I have a stone, it is rational to use the stone. It would be better, perhaps to use a hammer, but since I don't have a hammer, such is life. "The Hindoo does the best he candoo". Of course, the question remains, what is supposed to be the matter with induction that it needs justification? We don't feel that way about deduction. Why induction? What is it that induction needs that deduction apparently has?
By the way, Hume not only thought that we had no choice but to use induction in the sense that there was no better choice, he seems also to have thought that we literally could not choose not to use inductive inference since it was (to be anacronistic) hard-wired into us. So that is why we use induction (in the psychological sense of "why?" which asks for an explanation, and not a justification of our practice). And, as I said before, it was only the answer to that sense of "why?" that we could expect to get. The justification of induction, if there is one, was beyond the limits of human knowledge, just as Hume thought that it went beyond the limits of human reason to expect an answer to the question, do we know there is an external world. The justification of our belief in induction and the externiality of the world are the two mysteries which we can never expect to crack.