Called Complex For A(nother) Reason

Algebra Level 3

For n4n \geq 4, evaluate

i0!+i1!+i2!+i3!++in!.\large i^{0!}+i^{1!}+i^{2!}+i^{3!}+\cdots+i^{n!}.

Clarification:

  • i=1i = \sqrt{-1}.
  • n!=n×(n1)×(n2)××3×2×1n! = n\times(n-1)\times(n-2)\times\dots\times3\times2\times1.

Inspired by the series Called Complex For A Reason by Zandra Vinegar. Try her Part I, Part II and Part III
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