Question

13 If the equation ${\left(1+x+x^2+\dots ..+x^{17}\right)}^2-x^{17}=0$ 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 $k=1,2,3,4,\dots ..,34$ with $0<a_1<a_2<a_3<\dots \dots <a_{34}<1$ 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

Answer: 482

${\left(\frac{x^{18}-1}{x-1}\right)}^2=x^{17}$
$\Rightarrow {\left(x^{18}-1\right)}^2=x^{17}(x-1{)}^2\Rightarrow x^{36}-x^{19}-x^{17}+1=0\Rightarrow \left(x^{17}-1\right)\left(x^{19}-1\right)=0$
$\therefore x^{17}=1\text{ or }{x}^{19}=1$
$x=1^{1/17}\text{ or }x=1^{1/19}$
$Z=e^{\frac{2n\pi }{17}i}\text{ or }{e}^{\frac{2m\pi }{19}i}$
$\therefore a_1=\frac{1}{19};a_2=\frac{1}{17};a_3=\frac{2}{19};a_4=\frac{2}{17};a_5=\frac{3}{19}$
$\therefore \text{ sum }=\frac{159}{323}\Rightarrow m+n=482$