What is the next number of the series 7 16 8 27 9?

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 5 such that the first five 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.

For example,

if you want the next number to be 0, then use the rule:

U(n) = (109*n^5 - 1905*n^4 + 12405*n^3 - 37125*n^2 + 50276*n - 23340)/60 for n = 1, 2, 3, ...

if you want the next number to be 1, then use the rule:

U(n) = (219*n^5 - 3825*n^4 + 24895*n^3 - 74475*n^2 + 100826*n - 46800)/120 for n = 1, 2, 3, ...

if you want the next number to be 2, then use the rule:

U(n) = (11*n^5 - 192*n^4 + 1249*n^3 - 3735*n^2 + 5055*n - 2346)/12 for n = 1, 2, 3, ...

and so on.

Answer 2

40