Let `(a_1,a_2,a_3….,a_11)` be real numbers satsfying `a_1=15,27-2a_2 gt 0 ` and `a_k=2a_(k-1)-a_(k-2)" for " k=3,4…..,11`, If
`(a_1^2+a_2^2+……+a_11^2)/(11)=90` then the value of `(a_1+a_2+……+a_(1))/(11)`
is equal to __________.

Answer

0

Explanation

`a_(k) = 2a_(k -1) - a_(k -2)`
`rArr a_(1), a_(2),..., a_(11)` are in an AP.
`:. (a_(1)^(2) + a_(2)^(2) + ...+ a_(11)^(2))/(11) = (11a^(2) + 35 xx 11d^(2) + 10 ad)/(11) = 90`
`rArr 225 + 35 d^(2) + 150 d = 90`
`rArr 35 d^(2) + 150 d + 135 = 0 rArr d = -3, - (9)/(7)`
Given, `a_(2) lt (27)/(2)`
`:. d = -3 and d != -(9)/(7)`
`rArr (a_(1) + a_(2) + ...+ a_(11))/(11) = (11)/(2) [30 - 10 xx 3] = 0`