# I Feel Nostalgic!

Algebra Level 3

Let $$\{a,b\}$$ and $$\{m,n\}$$ be consecutive terms (in that order) of 2 different geometric progressions with common ratio $$r_1>0$$ and $$r_2>0$$ respectively, such that $(a-b)m=(a+b)n.$ Let $$z_1$$ and $$z_2$$ be $$2$$ complex numbers such that \begin{align} \arg(z_1z_2)&=\arctan(r_1)+\arctan(r_2)=\theta\quad\quad (0\leq \theta\leq \pi), \\ \arg\left(\frac{z_1}{z_2}\right)&=\frac{\pi}9. \end{align}

Find $$2\arg(z_1)$$.

Notation: The function $$\arg(z)$$ denotes the argument of a complex number.

Clarification: Take the value of $$\pi=3.1415$$.

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