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HMMT February

Algebra Difficulty 4.5 AIME Find the answer

Simplify: i0+i1+⋯+i2009i^{0}+i^{1}+\cdots+i^{2009}.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

By the geometric series formula, the sum is equal to i2010−1i−1=−2i−1=1+i\frac{i^{2010}-1}{i-1}=\frac{-2}{i-1}=1+i.

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