Called Complex For A(nother) Reason

Algebra Level 3

For \(n \geq 4\), evaluate

\[\large i^{0!}+i^{1!}+i^{2!}+i^{3!}+\cdots+i^{n!}.\]

Clarification:

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

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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