Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.0 AIME, harder Find the answer

2. Let $\mathrm{i}$ be the imaginary unit, simplify $(\mathrm{i}+1)^{2016}+(\mathrm{i}-1)^{2016}=$

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

$(i+1)^{2016}+(i-1)^{2016}=(2 \mathrm{i})^{1008}+(-2 \mathrm{i})^{1008}=2^{1009}$

Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.

$(i+1)^{2016}+(i-1)^{2016}=(2 \mathrm{i})^{1008}+(-2 \mathrm{i})^{1008}=2^{1009}$

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