Nth term of 2 8 16 26 38 52?

Answer 1

For the given sequence the simplest formula for the nth term is:

U{n} = n² + 3n - 2

However, that will only work if the sequence continues 68, 86, 106, 128, 152, ... (the difference between successive terms being the difference between the previous 2 terms plus 2).

If the sequence continues differently, then a different polynomial formula with higher powers will be required.

Answer 2

Any number that you choose can be the nth term. 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 in any of these ways - some simple, some complicated but all equally valid. 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.

Using the principle of Occam's razor, the simplest solution, based on a polynomial of degree 2,

U(n) = n^2 + 3n - 2