JEE Problem

Algebra Level 5

$\begin{cases} a_{1} = 15 \\ 27-2a_{2} > 0 \\ a_{k} = 2a_{k-1} - a_{k-2} & \text{for } k=3, 4,\ldots, 11 \end{cases}$

Let $a_{1}, a_{2}, \ldots , a_{11}$ be real numbers satisfying the conditions above. If $\dfrac{a_1^2 +a_2^2+a_3^2 +\cdots + a_{11}^2}{11} = 90$ , then find $\dfrac {a_1 +a_2+a_3 + \cdots + a_{11}}{11}$ .