To work out the ruke for each colour light, you have to investigate what colours light up for the numbers 1-30. After you have done this, you should have at least 3 for each colour. I have used a spreadsheet to do this. I have uploaded the one I used to show you how I did it. This helps, because it is easier to spot patterns in the numbers.
This is enough to work out the rule.
To work out the rule you have to work out the difference between two consecutive numbers in the sequence. This will give you the multiplier of 'n' for the nth term.
E.g. If the numbers that light up the red light are 2,7,12,17,22,27 the difference would be 5, so the start of the rule is 5n.
You then add or subtract, so that you get the first number in the sequence.
E.g. If you use the sequence in the first example, you would have to subtract 3 to make 2.
Lastly, to workout the 1st number that lights up all colours, you have to find the lowest common multiple. So, you have to list every number in each sequence, until you find a number that is in all 4 sequences.
Hope this helped and good luck!