in :
M1 = m.(DV/ve)
M1is the mass needed to psuh the useful mass to the desired velocity. But that fuel is not burned all at onnce and has to be pushed along. If you had to pushe the mass of fuel M1 all along, you'd have to push it with a mass : M2 = M1.(DV/ve)
But halfway through the burn, half of the mass M1 is gone, you don't need to push it all the way. So for short, we take the average betwee nothing to push and everything to push. What's hidden here is the area of a M1 by DV/ve triangle (while M1 is the area of a m by DV/ve rectangle).
One possible rigourous way to establish
(2.0) M2 = M1.(2DV/ve)
is to rewritte M1 = m.(DV/ve) as :
(4.0) M1 = m.(TWR.Dt/ISP)
Where Dt is the length of the burn (for example using ve = g.ISP and DV = g.TWR.Dt but an independant direct proof is possible), which can be generalized considering infinitesimal burn :
(4) dM1 = m.(TWR.dt/ISP)*
Which means to push a mass m during a laspse of time dt, you'll need dM1 as expressed.
To find (1) from (4), it suffices to sum over times varying from 0 to Dt = (ISP/TWR).(DV/ve), the duration of the burn.
(4) still holds when it comes to the second fuel mass carrying the first one, except, M1 varies with time :
(4') dM = m(t).(TWR.dt/ISP) , in particular : dM2 =M1(t).(TWR.dt/ISP)
Now if we make the reasonable assumption f a constant thrust,
M1(t) = M1(0)(1 - (t/Dt))
And summing for t in [0 ; 1] yield the factor 1/2 :
M2 = (1/2).M1.(TWR.Dt/ISP) = (1/2).M1.(DV/ve)
Which is (2.0).
Still working under the reasonable assumption that the thrust is constant the formula immediately extends to the subsequent fuel masses needed to lift M2, M3 and so on, and we have
(2) Mn+1 = (Mn/2).(DV/ve)
It has more to do with playing math than with playing KSP. I see it as a cute little paradox, or as an elaborate (false and) deceptive proof that 1+1 = 0.
As I write this I've realised where the error lies but won't give any clue, mostly out of tiredom after such a long post.
*Note that this equation is straightforward if you understand the definition of ISP, which is surprising given ISP was invented to circumvent translation issues of exhaust veocity between unit systems rather than to be something understandable.
