What is the number that completes the sequence 2 6 3 8 6 12?

Answer 1

First: no number completes the sequence since the implication is that there are no more numbers in the sequence.

Next, 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) = (2*n^5 - 35*n^4 + 230*n^3 - 699*n^2 + 966*n - 456)/4 for n = 1, 2, 3, ... and accordingly, the next term is 131.

Answer 2

First of all, the sequence if probably infinite and so it is not "completed" but simply extended. Next, there are infinitely many polynomials of order 6 that will give these as the first six numbers and any one of these could be "the" rule for determining the next number. 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.

One possible solution is based on a polynomial of order 5:

t(n) = (n^5 - 18*n^4 + 115*n^3 - 350*n^2 + 483*n - 228)/2 and accordingly, the next term is 131.