In number sequences, in order to determine what comes next, it is important to look into the differences between the adjacent numbers. In most cases, the pattern is immediately recognizable by going through a few of the numbers. However for this particular problem, it becomes clear that the whole sequence needs to be looked into closer.
That said these are the answers we get when we subtract the previous number from the succeeding one:
2nd - 1st = (-6) - 3 = (-9)
3rd - 2nd = 12 - (-6) = 12 + 6 = 18
4th - 3rd = 4 - 12 = (-8)
5th - 4th = 20 - 4 = 16
Looking further, we see that the answers have a pattern themselves which are:
(-9), 18, (-8), 16
We see that the negative numbers are increasing while the positive numbers are decreasing. Using this pattern, we can see that the pattern of the answers is:
(-9), 18, (-8), 16, (-7)
Going back to the original problem, what this means is that
6th - 5th = (-7) or
6th = (-7) + 5th = (-7) + 20 = 13
Therefore the next number in the pattern is 13.
One of the ways to find patterns in the sequence of numbers is to examine differences between the adjacent numbers. In this case, we would find that:
2nd - 1st = -6 - 3 = -9
3d - 2nd = 12 - (-6) = 18
Notice that second difference can be obtained from the first by multiplication by (-2).
4th - 3rd = 4 - 12 = -8. This is 1 greater than the first difference, -9.
5th - 4th = 20 - 4 = 16. This can be obtained from the previous difference by multiplication by (-2), again.
To continue in this fashion, the next difference, between the 6th and the 5th term, must be greater than -8 by 1. Since -8 + 1 = -7, the 6th term can be obtained by subtracting 7 from the 5th term, 20:
20 - 7 = 13
To get the next, 7th term, multiply (-7) by (-2) and add it to the 6th term: 13 + (-7)(-2) = 13 + 14 = 27.
So the next two numbers in the sequence would be 13 and 27. The longer sequence (check the following numbers using the same pattern) would be
3, -6, 12, 4, 20, 13, 27, 21, 33...
This is one of the possibilities of how the given sequence of numbers is formed. There might be others.
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