# Satisfying an inequality involving complex numbers

here's a question I haven't been able to solve.

Let $$z_{n}=(-\frac{3}{2}+\frac{\sqrt{3}}{2}i)^{n}$$. Find the least positive integer $$n$$ such that $$|z_{n+1}-z_{n}|^{2}>7000$$.

Ok so far I've done some really messy working using the addition formulas for $$|(\sqrt{3})^{n+1}(\cos\frac{5(n+1)\pi}{6}+i\sin\frac{5(n+1)\pi}{6})-(\sqrt{3})^{n}(\cos\frac{5n\pi}{6}+i\sin\frac{5n\pi}{6})|^{2}>7000$$ and I've gotten $$|(\sqrt{3})^{n}(-\frac{5}{2}\cos\frac{5n\pi}{6}-\frac{\sqrt{3}}{2}\sin\frac{5n\pi}{6}-i\frac{5}{2}\sin\frac{5n\pi}{6}+i\frac{\sqrt{3}}{2}\cos\frac{5n\pi}{6})|^{2}>7000$$, which looks kind of simpler I guess, if I didn't make any mistakes. But I'm not very sure how to continue or if this is not the right method because I'm just using brute force and I don't really see any trick anywhere. Could someone provide a hint? Thanks!

Let $$\alpha = -\frac{3}{2} + \frac{\sqrt{3}}{2}i$$. We have $$z_{n+1}-z_n = \alpha^n (\alpha-1)$$, and so $$|z_{n+1}-z_n| = |\alpha^n (\alpha-1)| = |\alpha|^n \cdot|\alpha - 1|$$, since multiplying complex numbers multiplies their modulus. You now solve $$|\alpha|^n \cdot|\alpha - 1| > \sqrt{7000}$$ which after computing $$|\alpha|$$ and $$|\alpha-1|$$, and then take the smallest integer solution, which happens to be $$n = \lfloor \log_{|\alpha|} \left( \frac{\sqrt{7000}}{|\alpha - 1|} \right) \rfloor + 1$$.
• @auscrypt Good answer and +1. It woos be better if you substitute $|\alpha|=3$ in your answer. – Qurultay May 21 at 15:37