Let $P\left(x\right)={(1+x+x^2+\mathrm{......}{x}^{17})}^2-x^{17}$ has 34 complex roots of the form $z_k=r_k{e}^{i2\pi a_k},k=1,2,\mathrm{3.....34}$ that $0<a_1\le a_2\le a_3\mathrm{......}\le a_{34}<1and{r}_k>0.$  If $a_1+a_2+\mathrm{......}{a}_5=\frac{m}{n}$ Where m,n are relatively prime integers then $m+n$ equals

$p\left(x\right)={\left(\frac{x^{18}-1}{x-1}\right)}^2-x^{17}=\frac{(x^{19}-1)(x^{17}-1)}{{(x-1)}^2}$

Now gcd(17,19)=1 and also p has 34 complex roots Hence all 16.17-th roots of unity And 18,19-th roots of unity are the root of p Then,  $z_m=e^{\frac{i2nt\pi }{19}},m=1,\mathrm{2.....18}$

$z_n=e^{\frac{i2m\pi }{17}},n=1,2,\mathrm{.......16} then{a}_k\in \{\frac{m}{19},\frac{n}{17}/m=1,\mathrm{.....18},n=1,\mathrm{........16}\}$