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Let \(a_{1},a_{2},a_{3}.....a_{11}\) be real numbers

satisfying \(a_{1}=15, 27-2a_{2}>0\) and

\(a_{k}=2a_{k-1}-a_{k-2}\) for \(k=3,4,5.....11.\)

If \(\frac{a^{2}_{1}+a^{2}_{2}+a^{2}_3.....+a^{2}_{11}}{11}=90\)

then the value of

\(\frac{a_{1}+a_{2}+a_{3}....a_{11}}{11}\) is equal to

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