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From the pattern (n + 4, n + 2, n + ?) I would say the next following number is 20 (n + 1).
n + 9
Any number that you choose can be the next logical number. It is easy to find a rule based on a polynomial of order 3 (a cubic) such that the first three 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 next number is 5 if you use the rule U(n) = -10*n^3 + 69*n^2 - 134*n + 77The next number is 11 if you use the rule U(n) = -9*n^3 + 69*n^2 - 123*n + 71The next number is 17 if you use the rule U(n) = -8*n^3 + 69*n^2 - 112*n + 65Intermediate values can be obtained with fractional coefficient.The next number is 5 if you use the rule U(n) = -9.833...*n^3 + 68*n^2 - 132.166...*n + 76 orU(n) = (-59*n^3 + 408*n^2 - 793*n + 456)/6
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.
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.The following rule:t(n) = 8.49861*n^6 - 195.47917*n^5 + 1758.46527*n^4 - 7843.68747*n^3 + 18098.03605*n^2 - 20183.83326*n + 8382 gives the next number is 14289.58, approx.