What is next number in series 3 5 8 13 22 with explanation?

Answer 1

Any number can be the next 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 and the next is the given number. There are also non-polynomial solutions.

The simplest polynomial solution, based on a polynomial of order 4 is

t(n) = (n^4 - 6*n^3 + 23*n^2 + 6*n + 48)/24 for n = 1, 2, 3, ...

Accordingly, t(6) = 38.

However, it is quite possible that these are based on a Fibonacci sequence defined by

t(1) = 3

t(2) = 5

and t(n+2) = t(n) + t(n+1) for n = 1, 2, 3, ...

If that is the case, then the next number is 35.

There is no way, other than reading the mind of the person who posed the question, as to which of these two, or the infinite alternatives is the correct answer.