How much does the market cap increase when a certain amount of $'s go in the market? Lets say that $1B worth of money is going in to bitcoin and its current market cap is $100B. How much would the market cap increase with that $1B?

Would it go to $101B, $105B, $110B, $100.1B?

I know that it will change according to how the money is put in. If they find someone selling by hand locally than it wouldn't affect the market price/market cap at all. 

But I'm more curious on the average of the increase? I'm guessing there is a formula for it in finance? What is it?
Given that the Bitcoin market is pretty unstable + unregulated I don't think it is possible to predict the outcome of this.

Have a look at the paper Price Manipulation in the Bitcoin Ecosystem by Neil Gandal et al. to get an idea of why this is very hard (if not impossible) to predict.