$21.$ A computer virus that spreads by way of e$-$mail messages is planted in $3$ machines the first day. Each day, each infected machine from the day before infects $5$ new machines. By the end of the second day, a software solution has been found to counteract the virus, and $1$ machine is clean at that point. Each day thereafter, $6$ times as many machines are clean as were clean the day before.

a$.$ Write a recurrence relation for the total number of infected machines on day n$.$

b$.$ Solve this recurrence relation.

c$.$ How many days will it be before the effects of the virus are completely gone?

$($The textbook says that T$($n$)=6$T$($n$-1)-6^$n$-2.6$T$($n$-1)$ is the number of infected machines and I was confused on why it was $6$ instead of $5.)$