Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.5 AIME Find the answer

Simplify: $i^{0}+i^{1}+\cdots+i^{2009}$ .

Your answer

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

Solution

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

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