A spaceship, at rest in a certain reference frame S, is given a speed increment of 0.50c.
Relative to its new rest frame, it is then given a further 0.50c increment. This process is
continued until its speed with respect to its original frame S exceeds 0.999c. How many
increments does this process require?
Velocity addition formula
(u + v)/(1 + uv/c^2)
The Attempt at a Solution
Speed after 1 increment = 0.50 c
Speed after 2 increments = (0.50 + 0.50) * c/(1 + 0.50*0.50)
= 0.80 c
Speed after 3 increments = (0.80 + 0.50)*c/(1 + 0.80*0.50)
= (13/14) c
But problem is I do not know how long to continue like this. Is there a simpler approach? I am not able to find if there is a generic expression for the velocity after nth increment for any given value of n.