f(1) = 1
f(2) = 2
f(3) = 3
f(4) = 4
f(5) = 5
[download]

But when you limit answers to functions having only natural numbers, of special kind (let that be polynoms with integral coefficients, with the largest power *no* more than three, then one and only one function fulfills the condition, and this is f(x)=x, and the correct answer is 6.)

Like everywhere else in mathematics, very many things depend on exact phrasing.

addition but I agree that OP has very badly phrased task

update 2+1 small typos

Consider the case of y = x + r*cos(pi*x) where r is any real, which produces a sine wiggle around a straight line, where the deviation from the straight line is maximum for half-values of y, but for integer values produces the illusion of y=x.

Update oic, you want to limit the types of functions - but without the business knowledge how can you do that? We know it's leasing, but without knowing the business, how do you know it doesn't have a risk component linked to a normal distribution based on the customer's age? That wouldn't be polynomial.

More update: yes fletch is right, s/cos/sin/ - my brain did the equivalent of a double negative when I was imagining this.

-M

Free your mind

ITYM sin not cos; the former intersects y=x for integers, while the later intersects for multiples of 0.5. And the extremums would be at n + (-1)ⁿ sin^-1( 1 / rπ )/π (for integer n).

</pedant> :)

Update: Gah, I had 2π rather than rπ because I used that as the constant when I checked things.

I think you understand my point (sorry for unperfect English) - sometimes tasks for finding a next number in some sequence of numbers are not useless, and such tasks help training ones brains.
OP has many flaws, and this is not really the sequence problem, and "Reverse engineering a formula" is a bad description of the problem

As a different note, our teacher said to us that the PI number contains, for example, encoded in it entire "War and Peace" by L.Tolstoy, starting from some position... you only need to find that N and enjoy the reading :)
While this is theoretically true, how can you benefit from this knowledge?
:) :)

With apologies for the thread drift:

Worse still, there are an uncountably infinite number of different formulas that can "justify" the sixth number being -7 x phi.

Surely a "number of different formulas" is an integer, and therefore countable?

You have a countable and finite number of intersections between a candidate function and the sample data, but the candidates may vary across real space in between the intersections, making them uncountable in number.

-M

Free your mind

What you say is certainly true in the abstract, but real systems, whether they be physical or financial, tend to follow relatively simple (albeit noisy) models. But I don't think the OP's system is even a real system; I think it's likely to be a very simple model of a system, probably involving exponential growth.

But all of this is guesswork. As others have said, the OP would probably get a great deal more help if only he would supply more data.