What is the next number in the series 86 73 58 41 26........?

Answer 1

One possibility given by Mehtamatics is:

t(n) = (n4 - 10n3 + 29n2 -110n + 606)/6

t(1..5) = {86, 73, 58, 41, 26} as required

→ t(6) = (64 - 10 × 63 + 29 × 62 - 110 × 6 + 606)/6 = 21

Another possibility:

t(2) = t(1) - 13

t(3) = t(2) - 15

t(4) = t(3) - 17

t(5) = t(4) - 15

Following this possible pattern, the next difference will be to subtract 13, giving:

t(6) = t(5) - 13 = 26 - 13 = 13.

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In fact you can have any number you want as the next number. Just specify the next number(s) you want and a polynomial formula (like the one Mehtamatics supplied) can be found to give the first 5 numbers as {86, 73, 58, 41, 26) and the number(s) you specify to follow.

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 rule isT(n) = (n4 - 10n3 + 29n2 -110n + 606)/6 for n = 1, 2, 3, ... and, accordingly the next tern is 21.