Multi Factorials!

Consider the following sequence n , ( n + 1 ) ! , ( n + 2 ) ! ! , n, (n+1)!, (n+2)!!, \ldots What positive integer n n makes the above sequence arithmetic progression ?


Note: Before you answer this problem, read this wiki and MathWorld to learn more about multifactorial.

3 2 6 1 4 There are infinitely many of them! 5

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Phillip Temple
Apr 4, 2018

The sequence in general is of the form (n+k)!!!....! where there are k factorials in the multifactorial.

So by definition of the multifactorial from mathworld, the terms of the sequence become:

n

(n + 1) * n * ... * 1

(n + 2) * n * ... * 1

(n + 3) * n * ... * 1

...

(n + k) * n * ... * 1

...

You can see an arithmetic progression in the first produc of each term, so the value of n has to end the multifactorial. Since all the factorials end when n = 1, n = 1!

When n = 1, the only items in each term are { n , n + 1 , n + 2 , ... n + k , ... } which is precisely an arithmetric progression we were looking for.

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...