Forget what these other guys said. Have you studied Emanuele Kant? He's a philosopher that says it's your duty as a citizen to make the world a better place if it's conveniently within your power to do so. How do you know this pervert isn't going to sleep with other students when she graduates? How do you know he's not taking advantage of his student-teacher relationship?
If you saw someone on the street get mugged, would you not do anything either just because it doesn't affect you in the grand scheme of things?
Go get me my finder's fee and report this guy.
At the very least you'll get some entertainment and maybe a free A.
So.. you think that they are having an intimate relationship and you think that he is giving her higher grades because of it...
You can report it, but if the professor has tenure then nothing will come of it unless you have actual proof. Only way I would even think of reporting it is if he graded on a curve and as a result it negatively impacts other people's grades. If she's failing assignments in her senior year, then eventually she'll have to deal with her incompetence when she going into the workforce.
Although, I am a big believer in "As long as it doesn't hurt anyone then it doesn't matter" but not everyone views the world the way I do, so do what you think is right.
Those are both likely outcomes if you report it. Nobody particularly cares about a lecturer sleeping with a student, but passing her because of it is a no-go. And as far as her, any academic dishonesty is typically going to result in failing the course as a minimum.
Would I do it? Nope. I really don't concern myself with other people, and I'm certainly not going to do something that could ruin two peoples' lives because I'm snippy I had to do more work in a class than somebody else.
Is there some kind of public safety concern if she passes this class on the back of a failed assignment? Absent that, worry about yourself.
First off, the OP only suspects that they are having an intimate relationship. He has no proof. Second, how is he a pervert? If he is sleeping with her, then he is sleeping with a consenting adult. Third, seeing someone get mugged and observing two people have an inappropriate intimate relationship are two VERY different things.
1. I told him to get proof in a previous post so I agree. If he does get the proof he should move forward.
2. It's a bit different than "they're both 18+ so they're consenting." I guess that's true but this is a guy in a position of power and influence over the other. Plus I'm pretty sure we're gonna be sexist here and say man is taking advantage of a young girl, just like if it were a male student and female teacher we'd all be like "niceee." (Lol at last sentence. This thread is fun.)
3. I agree that it's different but the point is just reporting things to make things better for others.
You need more proof than you've given so far to approach anyone about it, otherwise you are just creating trouble for them without justification.
First you said you 'think' they are in a relationship, then you said you are 100% certain. Either there is information being left out, or you don't actually have anything more than suspicion.
End point, you say you don't want anything bad to happen to the lecturer, (or her from the sound of it) other than the grades... If this is the case, I'd honestly just let it go. As long as her grades have no impact on yours, changing them won't do you any good. However, reporting it absolutely will result in his termination, possibly hers if it is true. Even if it isn't, it will very likely end badly for him at least.
TLDR: Unless you are absolutely sure, and you want to see them go down for it, let it lie. If you do want them to be corrected, make damn sure you have evidence before making a claim. Then go see the dean of students (or similar).