Consider the sequence


Form a new sequence, whose terms consist of the difference of the above sequence.


Repeat the process with the terms of this new sequence. When this is done sufficiently many times, you will eventually get the sequence

$$24, 24, 24, 24, \ldots$$

Why is this the case fundamentally, from a mathematical perspective? Why 24 exactly? I suspect it might have something to do with the fact that $24 = 4!$, though this could be completely off. Will it still work if we used the same algorithm with any arbitrary exponent $n$ instead of $4$? I can not answer this question because I can't seem to formalize this process in a way that allows me to see obvious convergence to a constant sequence.

  • 5
    $\begingroup$ if $f(x)$ is a polynomial of degree $n$ with leading coefficient $1$, then $f(x+1)-f(x)$ is a polynomial of degree $n-1$ with leading coefficient $n$. If you start from a polynomial of degree $n$ with leading coefficient $1$ and taking such a finite difference $n$ times. you get $n!$. $\endgroup$ Feb 4, 2018 at 6:59

1 Answer 1


This is the calculus of finite differences. You are starting with a function $f(x)$ (here $f(x)=x^4$) and defining a new one by $g(x)=f(x+1)-f(x)$, then iterating the procedure.

In general if $f(x)=a_nx^n+\cdots+a_0$ is a polynomial of degree $n$, then $g(x)=na_nx^{n-1}+\cdots$ is also a polynomial. Iterating then gives $n(n-1)a_n x^{n-2}+\cdots$ and after $n$ stages we get the constant $n(n-1)(n-2)\cdots 1a_n$.


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