Any number that you choose can be the next number. It is easy to find a rule based on a polynomial of order 7 such that the first seven 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, the rule:U(n) = (-31*n^6 + 693*n^5 - 5935*n^4 + 24315*n^3 - 48274*n^2 + 40272*n - 3360)/240 generates the above numbers for for n = 1, 2, 3, ... , 7 and n = 8 gives the next number as 206.
On what basis can this rule be described, in any way, as less correct than any other rule?
7. Your pattern is divide by 2, add 3, add 1.
7
7
What would be the next number in this series 15 12 13 10 11 8?
The pattern is - 5 then +2. 2 and 4 are the next numbers in the series.
The series 1 6 10 13 15 (1+5=6; 6+4=10, 10+3=13; 13+2=15) is completed by 15+1=16. The answer is 16.
This looks like a (subtract 5; add 2) series, so the next numbers are: 2, 4, -1, 1
24
What would be the next number in this series 15 12 13 10 11 8?
7
13
11
2
The pattern is - 5 then +2. 2 and 4 are the next numbers in the series.
The series 1 6 10 13 15 (1+5=6; 6+4=10, 10+3=13; 13+2=15) is completed by 15+1=16. The answer is 16.
6
21
Since each number is one less than the one before it, the next number would be 8.
24
This looks like a (subtract 5; add 2) series, so the next numbers are: 2, 4, -1, 1