Question

13 If the equation $\left(1+x+x^2+\ldots . .+x^{17}\right)^2-x^{17}=0$ has 34 complex root of the form of $Z _{ k }= r _{ k }\left(\cos 2 \pi a _{ k }+\sin 2 \pi a _{ k }\right)$, where $k =1,2,3,4, \ldots . ., 34$ with $0< a _1< a _2< a _3<\ldots \ldots< a _{34}<1$ and $r_k>0$. Also $a_1+a_2+a_3+a_4+a_5=\frac{m}{n}$, where $m$ and $n$ are coprime. Find the value of $(m+n)$.

Answer 1

$\left(\frac{x^{18}-1}{x-1}\right)^2=x^{17}$
$\Rightarrow \left( x ^{18}-1\right)^2= x ^{17}( x -1)^2 \Rightarrow x ^{36}- x ^{19}- x ^{17}+1=0 \Rightarrow\left( x ^{17}-1\right)\left( x ^{19}-1\right)=0 $
$\therefore x ^{17}=1 \text { or } x ^{19}=1$
$x =1^{1 / 17} \text { or } x =1^{1 / 19}$
$Z = e ^{\frac{2 n \pi }{17} i} \text { or } e ^{\frac{2 m \pi}{19} i}$
$\therefore a _1=\frac{1}{19} ; a _2=\frac{1}{17} ; a _3=\frac{2}{19} ; a _4=\frac{2}{17} ; a _5=\frac{3}{19} $
$\therefore \text { sum }=\frac{159}{323} \Rightarrow m + n =482 $