What number comes next 9 12 5 20?

Answer 1

As U{n} = (-47n⁴ + 598n³ - 2533n² + 4190n - 1992)/24

The next term U5 is (-47 × 5⁴ + 598 × 5³ - 2533 × 5² + 4190 × 5 - 1992)/24 = 42

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Alternatively, it may be two arithmetic sequences interlaced: 9, 5, -1, ... interlaced with 12, 20, 28, ..., ie the sequence is {9, 12, 5, 20, -1, 28, ...} making the next term -1.

In fact, you can have ANY number you want to come next. Give me a number and I'll give you a formula that gives that number, eg:

U{n} = (-45n⁴ + 514n³ - 2019n² + 3170n - 1512)/12

→U5 = (-45 × 5⁴ + 514 × 5³ - 2019 × 5² + 3170 × 5 - 1512)/12 = -1

Answer 2

According to Wittgenstein's Finite Rule Paradox every finite sequence of numbers can be a described in infinitely many ways and so can be continued any of these ways - some simple, some complicated but all equally valid. Conversely, it is possible to find a rule such that any number of your choice can be the next one.

The simplest polynomial solution is

U(n) = (16*n3 - 111*n2 +230*n - 108)/3 for n =1, 2, 3, ... and accordingly, the next term is 89.