Mathematics - Sequence & Series Question with Solution | TestHub

Let roots be ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_8$

$${\alpha }_1+{\alpha }_2+\dots +{\alpha }_8=4,{\alpha }_1{\alpha }_2..{\alpha }_8=\frac{1}{2^8}$$

∴ A.M. and G.M. of ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_8$ are equal.

$$\begin{array}{rlr}& \Rightarrow {\alpha }_1={\alpha }_2=\dots ={\alpha }_8=\frac{1}{2}& \Rightarrow a_2={}^8{C}_6\cdot {\left(\frac{1}{2}\right)}^6\cdot a_4={}^8{C}_4\cdot \frac{1}{2^4}\cdot a_6={}^8{C}_2\cdot \frac{1}{2^2}\end{array}$$