# MAA problem 1

Algebra Level 5

Let $${ a }_{ n }=1$$ for $$n=1$$ and $${ a }_{ n }=\left( \frac { 1 }{ 2 } +\frac { \sqrt { 3 } i }{ 2 } \right) { a }_{ n-1 }$$ for $$n>1$$. Now define the sum of the first $$k$$ terms of this series by $${S}_{k}$$. That is, $${ S }_{ k }=\sum _{ n=1 }^{ k }{ { a }_{ n } }$$. Determine the value of $$2{S}_{2015}$$.

Note: The $$i$$s in the answer choices are all outside the square roots, and $$i=\sqrt { -1 }$$.

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