Official solution
Given equation, x2 - 6x - 2 = 0
Roots are \[\alpha \] and \[\beta \].
So, \[\alpha + \beta = 6\] and \[\alpha \beta = - 2\]
In the question given, \[{a_n} = {\alpha ^n} - {\beta ^n}\]
\[\therefore\] \[{a_8} = {\alpha ^8} - {\beta ^8}\]
and \[{a_9} = {\alpha ^9} - {\beta ^9}\]
and \[{a_{{10}}} = {\alpha ^{{10}}} - {\beta ^{10}}\]
Now, the given equation
\[{{{a_{10}} - 2{a_8}} \over {2{a_9}}}\]
= \[{{{\alpha ^{10}} - {\beta ^{10}} - 2\left( {{\alpha ^8} - {\beta ^8}} \right)} \over {2\left( {{\alpha ^9} - {\beta ^9}} \right)}}\]
=\[{{{\alpha ^{10}} - {\beta ^{10}} + \alpha \beta \left( {{\alpha ^8} - {\beta ^8}} \right)} \over {2\left( {{\alpha ^9} - {\beta ^9}} \right)}}\] (as \[\alpha \beta = - 2\])
=\[{{{\alpha ^{10}} - {\beta ^{10}} + {\alpha ^9}\beta - \alpha {\beta ^9}} \over {2\left( {{\alpha ^9} - {\beta ^9}} \right)}}\]
= \[{{{\alpha ^9}\left( {\alpha + \beta } \right) - {\beta ^9}\left( {\alpha + \beta } \right)} \over {2\left( {{\alpha ^9} - {\beta ^9}} \right)}}\]
= \[{{\left( {\alpha + \beta } \right)\left( {{\alpha ^9} - {\beta ^9}} \right)} \over {2\left( {{\alpha ^9} - {\beta ^9}} \right)}}\]
= \[{{\left( {\alpha + \beta } \right)} \over 2}\]
= \[{6 \over 2}\] (as \[{ {\alpha + \beta } }\] = 6)
= 3