Mathematics - Binomial Theorem Question with Solution | TestHub

we know that $z^5-1=(z-1)\left(z-{\alpha }_1\right)\left(z-{\alpha }_2\right)\left(z-{\alpha }_3\right)\left(z-{\alpha }_4\right)$ Take $\log$ on both sides, diff.w.r.t. z and put $\mathrm{z}=2$. $\frac{5z^4}{z^5-1}=\frac{1}{z-1}+{\sum }_{i=1}^4\frac{1}{z-{\alpha }_i}$ At $z=2:\frac{5.16}{31}=1+{\sum }_{i=1}^4\frac{1}{2-{\alpha }_i}$ $\Rightarrow {\sum }_{i=1}^4\frac{1}{2-{\alpha }_i}=\frac{80}{31}-1=\frac{49}{31}$