$a_k=2a_{k-1}-a_{k-2}$


 $\Rightarrow a_1\text{, }{a}_2\text{,}\text{....}\text{, }{a}_{11}$ are in AP with let common difference be $\mathrm{d}$

 $\therefore \frac{a_1^2+a_2^2+\dots +a_{11}^2}{11}=\frac{11a^2+35\times 11{\text{d}}^2+110a\text{d}}{11}=90$

 $\Rightarrow 225+35{\text{d}}^2+150\text{d}=90$

    $35{\text{d}}^2+150\text{d}+135=0$

 $\Rightarrow \text{d}=-3\text{,}-\frac{9}{7}$

Given, $a_2<\frac{27}{2}\therefore \text{d}=-3 \text{ and }\text{d}\ne -\frac{9}{7}$

Hence, $a_1+a_2+.......a_{11}=\frac{11}{2}\left[2\times 15+10(-3)\right]=0$