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The value of ...

Question

The value of $100 \sum n = 0 i^{n!},$ where $i = \sqrt{- 1}$ is

A

97 + 2 i

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B

90 + 2 i

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C

95 + 2 i

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D

95 + i

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Solution

The correct option is C $95 + 2 i$
$100 \sum n = 0 i^{n!} = i^{0!} + i^{1!} + i^{2!} + i^{3!} + i^{4!} + i^{5!} + \dots + i^{100!} = i^{1} + i^{1} + i^{2} + i^{6} + i^{24} + i^{120} + \dots + i^{100 \cdot 99!} = i + i - 1 - 1 + (i^{4})^{6} + (i^{4})^{30} + \dots + (i^{4})^{25 \cdot 99!} = 2 i - 2 + {1 + 1 + \dots + 1 (97 times)} = 2 i - 2 + 97 = 95 + 2 i$

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