Multi Factorials!
Consider the following sequence What positive integer makes the above sequence arithmetic progression ?
Note: Before you answer this problem, read this wiki and MathWorld to learn more about multifactorial.
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1 solution
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.