There are infinitely many polynomials of order 5 that will give these as the first five numbers and any one of these could be "the" rule. 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.
A rule, based on a polynomial of order 4 is:U(n) = (35n^4 - 394n^3 + 1555n^2 - 2420n + 1284)/4 for n = 1, 2, 3, ...
174 =============== 18 = 15+3 54 = 18x3 57 = 54+3 171= 57x3 174= 171+3
The next number could be 26 The next number could be 12 - - - - - - - - - The next number that is in the sequence is 12.
6 12 9 18 15 30 27 54 51
the next two are 174 and 522.
The pattern is add three and then multiply by three. The next number is 174.
174 =============== 18 = 15+3 54 = 18x3 57 = 54+3 171= 57x3 174= 171+3
The next number could be 26 The next number could be 12 - - - - - - - - - The next number that is in the sequence is 12.
2 + 171=173 1 + 7 + 3 = 11 < 18
18
The next number in the sequence would be 25.
6 12 9 18 15 30 27 54 51
the next two are 174 and 522.
22
The pattern is add three and then multiply by three. The next number is 174.
The final number in this sum would be 18: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 = 171
The nest number in the sequence is 18. Note that the difference between each number and the next number in the sequence follows the simple sequence of 1,2,3,4. Obviously the next in the sequence of increases is 5, so 13+5=18.
What number best completes the sequence 1 1/2 5 10 1/2 18