Any number that you choose can be the next number. It is easy to find a rule based on a polynomial of order 6 such that the first six numbers are as listed in the question followed by the chosen next number. There are also non-polynomial solutions. Short of reading the mind of the person who posed the question, there is no way of determining which of the infinitely many solutions is the "correct" one.
The simplest solution, based on a polynomial of order 5, is
U(n) = (3*n^5 -50*n^4 + 315*n^3 - 910*n^2 + 1212*n - 540)/10 for n = 1, 2, 3, ...
Accordingly, the next term is 177.
12, 8
Hint: divide num and denom by their HCF... 18/27 = 2/3
-243
37
45
12, 8
81
15
45 54
21-3=18 18/2=9 9*3=27 that is the pattern so after the 36 it should be 33
The nth term is: n^2 +2 and so the next number will be 38
13 18 16 21 19 24 22 27 25 ... . This series consists of adding 5 to the first number and subtracting 2 from the next number, repeating the sequence in that order.
36
For each term, multiply the previous by -3. So, 27*-3=-81. -81 is the next term.
Hint: divide num and denom by their HCF... 18/27 = 2/3
243
25