The problem is that there are an infinite number of sequences that begin with these 5 numbers.

**The most likely solution sought is 13:**

Beginning with 3: subtract 9, add 18, subtract 8, add 16, subtract 7, add 14, etc. Each time, reduce the number subtracted by one, and double...

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The problem is that there are an infinite number of sequences that begin with these 5 numbers.

**The most likely solution sought is 13:**

Beginning with 3: subtract 9, add 18, subtract 8, add 16, subtract 7, add 14, etc. Each time, reduce the number subtracted by one, and double that number to add.

So, beginning with 3: 3-9 = -6, -6+2(9) = 12, 12-8 = 4, 4+2(8) = 20, 20-7 = 13, etc.

*****

Any five noncollinear points defines a unique quartic function. Here, if f(x) is the xth entry in the sequence, let

f(x) = 1/24(103x^4-1242x^3+5201x^2-8670x+4680)

then f(1) = 3, f(2) = -6 etc. Then the sixth number will be 213, followed by 839, etc.

*****

You do not have to define a way to get the next number in a sequence—you could select an arbitrary number, so any number is "correct" in that sense.