What is the 8th term for 12 59 294 1469 7344?

Answer 1

Any number that you choose can be the eighth term. It is easy to find a rule based on a polynomial of order 5 such that the first five numbers are as listed in the question followed by the chosen number in eighth position. 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 the following polynomial of order 4 U(n) = (376*n^4 - 3384*n^3 + 11186*n^2 - 15369*n + 7227)/3 for n = 1, 2, 3, ...gives U(8) = 135889.