JEE MAIN - Mathematics (2015 (Offline) - No. 4)
Let $$\alpha $$ and $$\beta $$ be the roots of equation $${x^2} - 6x - 2 = 0$$. If $${a_n} = {\alpha ^n} - {\beta ^n},$$ for $$n \ge 1,$$ then the value of $${{{a_{10}} - 2{a_8}} \over {2{a_9}}}$$ is equal to :
$$3$$
$$ - 3$$
$$6$$
$$ - 6$$
Explanation
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
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
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