What numbers come next 5 15 45 135?

Answer 1

Given any set of k number it is easy to find a rule based on a polynomial of order 4+k such that the first five numbers are as listed in the question and the next k are the given numbers. That is, it is easy to find a rule so that any number can be the "next" in the sequence.

So, suppose you want the next two numbers to be (5, 15) again then you define the rule as

Un = (24n5 - 400n4 + 2460n3 - 6890n2 + 8736n - 3915)/3

The simplest polynomial rule is

Un = (20n3 - 90n2 + 160n - 75)/3

and according to that, the next three numbers are 325, 655, 1165.

Alternatively, you could go for a power sequence

Un = 5*3n-1 which implies the next three terms are 405, 1215, 3645.

An order 5 polynomial, order 3 polynomial, geometric progression (plus others). There is no objective way to choose between them.