MAA problem 1

Algebra Level 4

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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