Which number comes next after this pattern 4-10-22-32?

Answer 1

Any number that you choose can be the next number. It is easy to find a rule based on a polynomial of order 4 such that the first four 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.

One possible rule, based on a polynomial of order 3 is

U(n) = (-4*n^3 + 33*n^2 - 53*n + 36)/3 for n = 1, 2, 3 and, accordingly, U(5) = 32.

But it is also possible to have

U(n) = (-4*n^4 + 36*n^3 - 107*n^2 + 147*n - 60)/3 which would give U(5) = 0

or

U(n) = (-31*n^4 + 278*n^3 - 821*n^2 + 1126*n - 456)/24 which would give U(5) = 1, and so on.

Answer 2

Any number that you choose can be the next number. It is easy to find a rule based on a polynomial of order 4 such that the first four 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.

One possible rule, based on a polynomial of order 3 is

U(n) = (-4*n^3 + 33*n^2 - 53*n + 36)/3 for n = 1, 2, 3 and, accordingly, U(5) = 32.

But it is also possible to have

U(n) = (-4*n^4 + 36*n^3 - 107*n^2 + 147*n - 60)/3 which would give U(5) = 0

or

U(n) = (-31*n^4 + 278*n^3 - 821*n^2 + 1126*n - 456)/24 which would give U(5) = 1, and so on.