There are infinitely many rules based on polynomials of order 5 such that the first five numbers are as listed in the question. 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 answer, based on a polynomial of order 2 (a quadratic) is
U(n) = (n^2 - n + 6)/2 for n = 1, 2, 3, ...
The next number in the sequence is 18. It's +1, +2, +3, +4, ....
18
18
18
18. 3+1=4, 4+2=6, 6+3=9, 9+4=13, 13+5=18.
The rule for this sequence appears to be adding consecutive prime numbers. The sequence starts with 3, then adds the next prime number 2 to get 4. It then adds 3 (the next prime number) to 4 to get 7, then 5 to 7 to get 12, and so on. Each number in the sequence is the sum of the previous number and the next prime number in order.
18 (3+1 4+2 6+3 9+4 13+5)
18
18
18 is the answer and here is why:- 3+1=4 4+2=6 6+3=9 9+4=13 13+5=18
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.
18 Add to previous number by 1 then 2 then 3 then 4 and 5 3+1=4 4+2=6 6+3=9 9+4=13 so then 13+5=18