Bacteria Problem

I have found a interesting question

Question:

A certain bacteria is known to reproduce asexually two new bacteria every minute under a certain growth condition, On average, 1 out of 4 newly reproduced bacteria cannot reproduce themselves. After a significant time has passed, what would be the percentage of reproducible bacteria within the population?

After consideration, the solution came out:

First of all, lets declare some notations:

: non-reproductive bacteria increased at time t(non-reproductive bacteria)

: total reproductive bacteria at time t

: initial quantity of bacteria

Thus, the relationships below are gained:

due to Law of Large Numbers.

(initial quantity)

After that, we can get a geometric series

Then, we need to find the sum to t

Hence: