How the following Algebraic expression came?

Question

I am a Molecular Biologist and working on Mathematical Modeling for my PhD project. I had been studying model and I could follow uptill to P3. But I could not get about extreme right of the equation when it was abstracted to Pn. I can't understand, how 'r' appears in the denominator or what possible method was used for this particular abstraction. I shall be grateful if I get some assistance from experts. Best Regards.

Answer 1

This is an example of a geometric series. The general format is $$1 + s + s^2 + s^3 + … + s^n = \frac{1-s^{n+1}}{1-s} $$ which you can verify by writing it as $$(1-s)(1 + s + s^2 + s^3 + … + s^n) = 1-s^{n+1} $$ and carrying out the multiplication on the left hand side and watching almost all the terms merrily cancel away until all that is left is the right hand side.

For your problem, set $s=1-r$.