Mock AIME 1 2007-2008 Problems/Problem 5

Problem 5

Let $S = (1+i)^{17} - (1-i)^{17}$ , where $i=\sqrt{-1}$ . Find $|S|$ .

Solution

Rewriting the complex numbers in polar notation form, $1+i = \sqrt{2}\,\text{cis}\,\frac{\pi}{4}$ and $1-i = \sqrt{2}\,\text{cis}\,-\frac{\pi}{4}$ , where $\text{cis}\,\theta = \cos \theta + i\sin \theta$ . By De Moivre's Theorem, $\begin{align*} \left(\sqrt{2}\,\text{cis}\,\frac{\pi}{4}\right)^{17} - \left(\sqrt{2}\,\text{cis}\,-\frac{\pi}{4}\right)^{17} &= 2^{17/2}\,\left(\text{cis}\,\frac{17\pi}{4}\right) - 2^{17/2}\,\left(\text{cis}\,-\frac{17\pi}{4}\right) \\ &= 2^{17/2}\left[\text{cis}\left(\frac{\pi}{4}\right) - \text{cis}\left(-\frac{\pi}{4}\right)\right] \\ &= 2^{17/2}\left(2i\sin \frac{\pi}{4}\right) \\ &= 2^{17/2} \cdot 2 \cdot 2^{-1/2}i = 2^9i = \boxed{512}\,i \end{align*}$

Mock AIME 1 2007-2008 (Problems, Source)
Preceded by Problem 4	Followed by Problem 6
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Mock AIME 1 2007-2008 Problems/Problem 5

Problem 5

Solution

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