How do you write a formula for the nth term?

Answer 1

A question about the nth term of a sequence, which does not give you the formula for the nth term, can only contain a finite number of terms. The issue then is to find a function that will generate these numbers. Now, given any set of n numbers, it is always possible to find infinitely many polynomials of order n such that the polynomial is the formula for the nth term. Furthermore, there will be at least one polynomial of order (n-1) that will meet the requirements. There will be non-polynomial functions as well. 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, given the sequence 1, 3, 5, 7, 9, 11, ... what do you think is the formula for the nth term?

It could be t(n) = 2n - 1

But, it could also be

u(n) = (n^6 - 21*n^5 + 175*n^4 - 735*n^3 + 1624*n^2 - 1524*n + 600)/120, or any of an infinite number of 6th degree polynomials in n.

The next terms for these two formulae would be t(7) = 13 and u(7) = 19.

But you were not given the 7th term so you do not know which formula is correct! If the 7th term is given, you can find infinitely many 7th degree polynomials that will work.

Then the only solution is Occam's razor: go for the answer that looks simplest. Or go for one that you like!