One trained person can train three people in 3 days and those three people individually can train three more people in 3 days. If the process goes on like this and initially there is only one trained person in the company, then how many days will it take to get 1093 trained people in all, given that one person trains only three people?

The number of newly trained people after the first three day period is 3.
Including the initial member who trained them, the total number of trained people would be 1+3

The number of newly trained people after the second trained people is 9.
Including the number of people who are already trained, we get the total number of trained people to be 1+3+9

We can follow through with this reasoning and see that the total number of trained people make the sum of a GP with the common ratio 3 and the number of terms equal to the number of three day periods required. 
We want the total number of trained people to be 1093. 
Using the formula for the sum of an GP with n terms we get, $$\frac{a\left(r^n-1\right)}{r-1}=\frac{1\left(3^n-1\right)}{3-1}=1093$$
$$3^n-1=2186$$
$$3^n=2187$$
from here, we can keep checking the value of n and we find that n=6 satisfies the equation. 

So there are a total of 6 three day periods required, hence, a total of 18 days required for 1093 people to be trained.