What is the nth term of -4 -1 2 5 8?

Answer 1

Any number that you choose can be the nth number. 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. 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.

For example, the position-to-value rule given by

U(n) = (-7*n^5 + 105*n^4 - 595*n^3 + 1575*n^2 - 1558*n - 480)/120 is a possible solution for the given set of numbers.

A simpler solution is V(n) = 3*n - 11, but that does not make it any more correct than the first one.

Answer 2

According to Wittgenstein's Finite Rule Paradox every finite sequence of numbers can be a described in infinitely many ways and so can be continued any of these ways - some simple, some complicated but all equally valid.
The simplest rule which will fit the two given numbers is T(n) = -1254*n + 1250 for n = 1, 2, 3, ...

Answer 3

3n -7