QJEEAdv10Ma877d: JEE Advanced 2010 Mathematics - Class 11 JEEAdv10Ma877d Solution: Let ......, be real numbers satisfying . if , then the value of …

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

$\Rightarrow {a_1},{a_2},\,\,.....,\,\,{a_{11}}$ are in AP

$\therefore$ ${{a_1^2 + a_2^2 + \,\,....\,\, + \,\,a_{11}^2} \over {11}}$

$= {{11{a^2} + 35 \times 11{d^2} + 10ad} \over {11}} = 90$

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

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

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

Given, {a_2} < {{22} \over 7} $\therefore$ $d = - 3$ and $d \ne - {9 \over 7}$

$\Rightarrow {{{a_1} + {a_2} + \,\,...\,\, + \,\,{a_{11}}} \over {11}}$

$= {{11} \over 2}[30 - 10 \times 3] = 0$