What is the next number 88 44 22 11?

Answer 1

What number would you like to be next?

Using U{n} = (5.25n⁴ - 58n³ + 249.75n² - 554n + 621)/3

it gives {88, 44, 22, 11} for n = 1, 2, 3, 4, so the next term is U{5} = 42.

However, the solution I suspect your teacher wants is that it is a GP with each term half the previous term (ie a common ratio of ½) so the next term is 11×½ = 11÷2 = 5½ = 5.5

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.

For example, using the simplest polynomial rule

T(n) = (-11n3 + 132n2 - 583n + 990)/6 gives the next term as 0, while another,

T(n) = (9n4 - 112n3 + 579n2 - 1616n + 2196)/12 gives the next term as 18.