The King's Guard
In a kingdom, a king's guard is deployed in a special formation. The number of guards in each row is determined by a power of \(3\). In the first row, there are 3 guards, in the second row, there are 9, in the third, there are 27, and so on. The king needs to know the total number of guards in different formations.
(i) How many guards will be in the 9th row?
(ii) If a few new formations require \(3^{10}\) guards in a single row, which row number is this?
(iii) The number of guards in a different formation doubles every \(2\) days, starting with \(12\) guards. How many guards will there be after \(6\) days?
(iv) What is the ratio of the number of guards in the \(8\)th row to the number of guards in the \(4\)th row?