Mathematics - Sequence & Series Question with Solution | TestHub

$$\begin{array}{rl}& a_k=2a_{k-1}-a_{k-2}\\ & \Rightarrow a_1,a_2,\dots ,a_{11}\text{ are in AP }\\ & \therefore \frac{a_1^2+a_2^2+\dots +a_{11}^2}{11}\\ & =\frac{11a^2+35\times 11d^2+10ad}{11}=90\\ & \Rightarrow 225+35d^2+150d=90\\ & 35d^2+150d+135=0\\ & \Rightarrow d=-3,-\frac{9}{7}\end{array}$$ Given, $a_2<\frac{27}{2}\therefore d=-3$ and $d\ne -\frac{9}{7}$ $$\Rightarrow \frac{a_1+a_2+\dots +a_{11}}{11}$$ $$=\frac{11}{2}[30-10\times 3]=0$$