Mathematically an arithmetic progression is a sequence of numbers where the difference between consecutive terms is constant e.g. 2, 4, 6, 8, 10...
A geometric progression is one where each term is in a common ratio to the one preceeding it e.g 2, 4, 8, 16, 32, 64 ... is a geometric progression where 2/4 = 4/8 = 8/16 = 16/32 ... = 0.5 is the common ratio.
How can one apply this maths to painting a grey scale? The obvious way is to apply the progression to the amount of black paint in the mix. However, if you had the expertise you could attempt to apply the progression to the value (luminance) of the mixes instead. Maybe this is what is required for the third scale, rather than progress by arithmetic of geometric addition of amounts of black paint, paint a scale with even steps in value.
Say we are going to attempt 10 step progressions with different amounts of black paint (effectively exploring the tinting power of the black).
An arithmetic (linear) approach would be, listing the parts of white to black in each mix 
White:Black
10 : 0
9 : 1
8 : 2
7 : 3
...
1 : 9
0 : 10
A geometric approach (amount of black increasing geometrically) would be to take an amount A of white and add a unit of black, then take amount A of the resulting mix1 and add a unit of black to make mix 2, then take amount A of mix2 and add a unit of black etc. To be truly geometric it is important to keep ratio of the amount of base mix to black added each time the same. Marigold didn't mention that detail, and without it the results will be slightly different. The
common ratio between mixtures is then a constant A/(A+1)
The resulting scale depends on the relationship between amount "A" and the "unit" of black. Black paint is usually strong, so with A of 1 unit the white would be overwhelmed within a couple of steps and the mix look black.
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Alas, I have no idea how to progress geometrically. If I start with two drops on the second segment I go up to hundreds of drops in the last.

Actually that is the same as taking A = 1 unit, mix 1 is half and half black and white, mix 2 is 1/4 white, mix 3 is 1/8 etc. by 1/64 (1.56%) white, if not before, it looks black. But progressive mixing, in this case half and half each time, is easier way to think about it than counting out hundreds of drops
Just dawned on me to ask what medium are you using, you talk of "drops" so some kind of liquid colour I guess. I have been thinking in mixing dollups of acrylic or oils, but what I have said about amounts would also work for making dilute solutions of a liquid black.
Anyway say we do some progressive mixing with A of 5 units, mix 1 is 5/6 (83.3%) white, mix 2 is 25/36 (69.4%) white etc. The fractions get a bit boggling, but if I list the percentage whites for each mix using different amounts A and adding a unit of black each time it might show something of
how it works.
Mix A = 2 A = 5 A = 10
1 66.6% 83.3% 90.9%
2 44.4% 69.4% 82.6%
3 29.6% 57.8% 75.1%
4 19.8% 48.2% 68.3%
5 13.1% 40.2% 62.1%
6 8.8% 33.5% 56.5%
7 5.8% 27.9% 51.3%
8 3.9% 23.3% 44.7%
9 2.9% 19.3% 42.4%
Starting with 10 times as much white as black (A = 10 units), after 9 mixing steps you would still have a good amount of white in the mix, maybe too much.
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Using paint (not the MS variant) how do I make a 10 segment scale from white to black progressing geometrically?

There are many different geometric series that go from somewhere above 0 to somewhere below 100 in 9 steps. An arbitrary example would be
28.7, 33.0, 37.9, 43.5, 50.0, 57.4, 66.0, 75.8, 87.1 (common ratio 1.148). In your paint programme set the value/luminance colour coordinate to these values. Not sure how meaningful that would be, other than to give a feel for values being in common ratio rather than linear  something different from what we have been doing mixing amounts of paint that way. I suspect that mixing in a geometric progression of amounts may teach us something about tinting power, not sure that seeing a geometric progression of values does.
But curiosity got the better of me, so here is a comparison of (first row) a linear variation of CIE Lab L values in steps of 10, along side two arbitrary geometric progressions of L. Both have too big a leap to black at the end.
But perhaps the bottom one is a bit like the top? If a progression is chosen that covers the dark greys nearer to black then there are big steps in the light grey area instead.
Hope this is of some help.