You have some good answers already but I would like to add one point.
First, as some others have pointed out, your formula is wrong.
Not: F = 440 + 2^(n/12)
It should be: F = 440 x 2^(n/12)
Consider the case n = 12 (an octave). Your formula would suggest that 2Hz is added but in fact the frequency is doubled (times 2). An octave up from 440Hz is 880Hz not 442Hz.
Now my main point: you have the cart before the horse. No one sat down and devised that weird formula before they first started to sing. It was developed long after and, as some have mentioned, it only applies to the relatively modern equal temperament system. Older systems did not use strange irrational numbers.
The answer, with the horse before the cart, is that this happens to be how our ears work. We don't hear 440Hz, 880Hz, 1320Hz, 1760Hz, etc as equal steps. It would sound like decreasing steps: an octave (A to A), a fifth (A to E), a fourth (E to A); roughly, ignoring precise temperament considerations.
Others have touched on why so I won't go into that but if you want to convince yourself that it is so then try this experiment.
Find a man and a woman who can sing simple tunes reasonably well but are neither trained musicians nor mathematicians or physicists. Hence they cannot read music and do not know that formula.
Give the woman a 440Hz (A4) note to start and ask her to sing Twinkle, Twinkle Little Star. Her first two notes will be (approximately) 440Hz and then she will hopefully jump to 660Hz (E5). 220Hz up or 1.5 times the frequency.
Now ask the man to sing with her. Unless he has an exceptionally high voice, he won't sing the same notes but an octave down. So, he will start on 220Hz (A3) and then he will jump to 330Hz (E4). 110Hz up and 1.5 times the frequency. So, what sounds like the same change is still 1.5 times the frequency but a smaller step of 110Hz.
This also shows that moving an octave (doubling or halving the frequency) is perceived as pretty much the same note. If the woman was close to 440Hz and the man close to 220Hz, they would be considered as in tune. Others have touched on why: all of the harmonics of the woman's 440Hz would also be harmonics of the man's 220Hz.
This is common in physics. Look at the formula for the gravitational attraction between two bodies. Newton did not invent that formula and impose it on the Solar System. He figured out that it did a very good job of explaining and predicting the Solar System's behaviour. Your formula is similar: it was not imposed on music. Music came first and the formula was developed as a model for it later.