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Imaginary i

Source: 1964 AHSME Problem 34

August 26, 2013
AMC

Problem Statement

If nn is a multiple of 44, the sum s=1+2i+3i2+...+(n+1)ins=1+2i+3i^2+ ... +(n+1)i^{n}, where i=−1i=\sqrt{-1}, equals:
<spanclass=′latex−bold′>(A)</span> 1+i<spanclass=′latex−bold′>(B)</span> 12(n+2)<spanclass=′latex−bold′>(C)</span> 12(n+2−ni) <span class='latex-bold'>(A)</span>\ 1+i\qquad<span class='latex-bold'>(B)</span>\ \frac{1}{2}(n+2) \qquad<span class='latex-bold'>(C)</span>\ \frac{1}{2}(n+2-ni) \qquad
<spanclass=′latex−bold′>(D)</span> 12[(n+1)(1−i)+2]<spanclass=′latex−bold′>(E)</span> 18(n2+8−4ni) <span class='latex-bold'>(D)</span>\ \frac{1}{2}[(n+1)(1-i)+2]\qquad<span class='latex-bold'>(E)</span>\ \frac{1}{8}(n^2+8-4ni)
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