If α and β are the roots of the quadratic equation \x^2-6\x-2=0 and if \a_\n=\\alpha^\n-\\beta^\n then the value of \frac{\a_10-2a_\8}{2\a_9}

NTSE Stage 2 SAT 2016 Question Paper

Question : 45 of 100

Marks: +1, -0

If α and β are the roots of the quadratic equation

\x^2-6\x-2=0

and if

\a_\n=\α^\n-\β^\n

then the value of

{\a_10-2a_\8}/{2\a_9}

Solution:

\x^2-6\x-2=0

\α^2-2=6\α

\β^2-2=6\β

\α+\β=6,\α\β=-2

\d_\n=\α^\n-\β^\n

{\a_10-2\a_8}/{2\a_9}={\α^10-\β^10-2(\α^8-\β^8)}/{2(\α^9-\β^9)}

{\α^10-\β^10+\α\β(\α^8-\β^8)}/{2(\α^9-\β^9)}

{\α^10+\α^9\β-(\α\β^9+\β^10)}/{2(\α^9-\β^9)}

{\α^9(\α+\β)-\β^9(\α+\β)}/{2(\α^9-\β^9)}

{(\α+\β)(\α^9-\β^9)}/{2(\α^9-\β^9)}

6/2=3

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