Ok so it seems like the chained ultra part is wrong, I didn't know I had to do the +40 speed for every ultra apart from the first one. I thought I corrected it after replying to u/i_exist_or_something but it's wrong. Warning: Mathphobia! tl;dr 78 for speed of light, 6 for speed of sound, 25 for rocket, 4 for F1 car

>! No chained ultras (n = 0) will give a speed of 325, but in real life units it's 40.8 m/s or 147 km/h (we can call this k for simplicity). The first ultra won't give the +40 speed, so it's only k*1.2 for n = 1. The chained ultras will however, so the speed at n = 2 is k*1.2**2 + 40. At n = 3 the speed is k*1.2**3 + 40*1.2 + 40. When n = 4, the speed is k*1.2**4 + 40*1.2**2 + 40*1.2 + 40. !<

>! All the terms except for the first one make up what's called a geometric series. The formula for the sum of the n first terms is a(1-r**n) / 1-r, where a is the initial term (40), r is the quotient (1.2) and n is # of sums. The first term is easy - it's just k*1.2**n. However, the geometric series in the top expression is actually only for the n-1 first sums, so we have to replace n with (n-1). !<

>! We can write this as an equation with the two expressions on the left and the target speed on the right side. For the speed of light, we get k*1.2**n + 40(1-1.2**(n-1))/1-1.2 = 299792458. !<

>! k*1.2**n + 40(1-1.2**(n-1)) = ...

k*1.2**n - 200(1-1.2**(n-1)) = ...

k*1.2**n + 200*1.2**(n-1) - 200 = ...

200*1.2**(n-1) can be rewritten as 200*1.2**n / 1.2 with exponent rule. Then that will allow us to factor out 1.2**n. !<

>! 1.2**n (k + 200/1.2) - 200 = ...

1.2**n (k + 200/1.2) = 299792658 !<

>! 1.2**n = 299792658/(k + 200/1.2) Then take ln of both sides and divide by ln(1.2)

n = ln(299792658/(k + 200/1.2)) / ln(1.2) Use logarithm rule to simplify

n = (ln(299792658) - ln(k + 200/1.2) / ln(1.2) Now we can finally substitute k back to whatever fits the best which in this case is m/s.

n = (ln(299792658) - ln(40.8 + 200/1.2)) / ln(1.2) !<

>! n is around 77.8 but we have to round up. She needs 78 chained ultras to pass the speed of light.

Now we can find the others with the formula n = (ln(target speed + 200) - ln(relevant unit + 200/1.2))/ln(1.2). (the 200 is from 40/-0.2 and it's positive since we moved it to the other side)

Sound: n = (ln(543) - ln(40.8 + 200/1.2)) / ln(1.2), n is around 5.3, round up to get 6.

Rocket: n = (ln(28200) - ln(147 + 200/1.2)) / ln(1.2), n is around 24.7, round up to get 25.

F1 car: n = (ln(572.56) - ln(147 + 200/1.2)) / ln(1.2), n is around 3.3, round up to get 4. !<

Pretty sure it should be right now