<s>[INST] Find the value of $ \frac{ arg(z_1) +arg(z_2) + \cdots + arg(z_{25})}{25}$ 
If the equation $(1+x+x^2+x^3+\cdots x^{17})^2-x^{17}=0 $ has roots $z_1,z_2,\cdots ,z_{34}$ and $0 \lt arg(z_1) \lt arg(z_2) \lt \cdots \lt arg(z_{34}) $ then find 
  $$ \frac{ arg(z_1) +arg(z_2) + \cdots + arg(z_{25})}{25}$$

$$(x^{18}-1)^2-x^{17}(x-1)^2=0$$
$$(x^{17}-1)(x^{19}-1)=0$$
Neglecting $x=1$ as it came out to be extra 
roots .
$$arg \rightarrow \frac{2\pi}{19},\frac{4\pi}{19},\frac{6\pi}{19},\cdots ,\frac{36\pi}{19}$$
$$arg \rightarrow \frac{2\pi}{17},\frac{4\pi}{17},\frac{6\pi}{17},\cdots ,\frac{32\pi}{17}$$
Now  I am stuck at predicting first 25 $arg$ satisfying inequality 
 [/INST] There is a bug in this problem. Roots are 34 but only 16 have positive arguments: after all they are roots of the unity and are all around the unit circle.
Here we have the 34 arguments sorted ($\pi$ omitted to keep the list shorter)
$-\dfrac{18}{19},\;-\dfrac{16}{17},\;-\dfrac{16}{19},\;-\dfrac{14}{17},\;-\dfrac{14}{19},\;-\dfrac{12}{17},\;-\dfrac{12}{19},\;-\dfrac{10}{17},\;-\dfrac{10}{19},\;-\dfrac{8}{17},\;-\dfrac{8}{19},\;-\dfrac{6}{17},\;-\dfrac{6}{19},\;-\dfrac{4}{17},\;-\dfrac{4}{19},\;-\dfrac{2}{17},\;-\dfrac{2}{19},\;0,\;0,\;\dfrac{2}{19},\;\dfrac{2}{17},\;\dfrac{4}{19},\;\dfrac{4}{17},\;\dfrac{6}{19},\;\dfrac{6}{17},\;\dfrac{8}{19},\;\dfrac{8}{17},\;\dfrac{10}{19},\;\dfrac{10}{17},\;\dfrac{12}{19},\;\dfrac{12}{17},\;\dfrac{14}{19},\;\dfrac{14}{17},\;\dfrac{16}{19},\;\dfrac{16}{17},\;\dfrac{18}{19}$
If we consider the arithmetic mean of these positive result, we get $\dfrac{6058 \pi }{8075}\approx 2.357$
I don't know
I hope this helps
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