A university runs computer labs with many computers. Each day each computer is in one of two states: It is functioning, or it is shut down and requires repair. Each day a staff member inspects all functioning computers. It has been observed that after the inspection, 90% of the functioning computers will remain functioning and the remaining 10% will be shut down and require repair, which will be performed the next day or later. Each day another staff member tries to repair the computers that were shut down on the previous day or earlier. A fraction of c of these computers will be repaired during the day and will be functioning the next day, while the remaining fraction 1 c of them will remain shut down the next day. Here c is a constant, independent of time, such that 0 < c < 1. The university requires that 85% of all computers are functioning each day. You can consider this as a requirement for the steady state of the dynamical system described above. Find the value of c that satisfies the university requirement, that is, 1 what fraction of broken computers have to be repaired every day. This is a simple linear algebra problem that can be solved by hand.

This is what I got so far: let f_n and r_n denote the fractions of functioning and broken computers, respectively, on day n. Then we have the following recursive relationships: f_n+1 = 0.9 f_n + (c) r_n and r_n+1= 0.1 f_n + (1-c) r_n We want to find the value of c that leads to a steady state where 85% of computers are functioning, i.e., F_n = 0.85 for all n. please help me with this question.

PLEASE DON'T COPY FROM OTHER ANSWERS.