I've read a grammar rule. If you use a preposition and then a verb, it must end with "ing"
Is it always true, or there are exceptions?
Learn English – Is it true that if you use a preposition and then a verb, it must end with “ing”
grammaticalityprepositionssuffixesverbs
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I’ve heard of it, but never heard it. The absence of a direct object after a phrasal verb that normally has one is not unprecedented. FF has already mentioned come after, but there are other examples. You can get on or get off a bus, or you can just get on or get off. You can go without dinner, or you can just go without. I don’t imagine anyone objects to those now, if they ever did. I suspect the only thing intransitive come with can be charged with is being new.
I wouldn’t be surprised if, as Lunivore says, it’s a construction found in Afrikaans. German has the cognate separable verb mitkommen (although admittedly that occurs only intransitively).
If you provide for [someone/something in need] without specifying exactly what, it means you make available to them whatever they need - often food (provisions/provender). You can also provide for [some eventuality] without specifying what, meaning you have plans/resources to deal with that eventuality should it come to pass. Both these usages have been around a long time, and always involve for, not to.
To provide [something] to [someone] is a far more recent usage...
Per @JeffSahol's answer, provide X to Y often implies that Y did actually receive X, whereas provide X for Y can be used even if Y doesn't avail himself of the X which is on offer. But often it's an idiomatic choice where people repeat the version they hear most. Other than that, the modern trend towards to clearly disambiguates from the first two usages given above.
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This is kind of tricky, let's see if I can explain...
As Claudiu pointed out already, gerund is the term. However, expanding on what he said, "a verb ending in -ing" is not necessarily a gerund; it could be a present participle. A gerund is an -ing form used as a noun (more precisely, as the head of a noun clause; it might still behave as a verb within the clause itself).
Thus, the rule you read is kind of okay, yet it is also kind of backwards. Being able to be preceded by a preposition is a nominal characteristic, not a verbal one. You can't really put a preposition in front of a verb — once you do that it needs to start functioning as a noun, and "normal" verbs can't do that. So you need a verb form that works as a noun. And that would be the gerund.
But what about this, you say:
Well, yeah, the verb fail somehow managed to start working as a noun all by itself. But again, the whole point is that when it acts as a noun it is, well, a noun; it is no longer a verb.