Factorials with exponents. Is their a way?

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I know of multiplication factorials with the 4! = 4*3*2*1 and I know of the addition with the $n$'th triangle.

I am busy deriving my own equation for something, and I am getting stuck on how to furthur my progress.

I need to have a similiar factorial for exponents, but I have no idea how to simply this.

Equation based on financing:

If $n = 1, F = P(i)$

If $n=2, F = P(i^2 +i)$

If $n =3, F = P(i^3 + i^2 + i)$

You can see where I am going with this. Can anyone help me simplify this, because $n$ could be a large number and that is a lot to write?

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You have a geometric sum. The $n$th term is given by

$$\sum_{m=1}^n i^m =\frac{i-i^{n+1}}{1-i}=i\frac{1-i^n}{1-i}$$