Let α and β be the roots of equation $$X2$$ $$-$$ $$6x$$ $$-$$ $$2$$ $$=$$ O. If $$an=$$αn−βn, for n≥$$1$$,then the value $$a10$$−$$2a82a9$$ as is equal to

Question

Let  α and β be the roots of equation $$X2$$ $$-$$ $$6x$$ $$-$$ $$2$$ $$=$$ O. If $$an=$$αn−βn, for n≥$$1$$,then the value $$a10$$−$$2a82a9$$ as is equal to

Infinity Learn · Accepted Answer

Given,α and β are the roots of the $$equationx2$$−$$6x$$−$$2=0$$∵$$an=$$αn−βn for n≥$$1$$∴ $$a10=$$α$$10$$−β$$10$$ $$a8=$$α$$8$$−β$$8$$⇒$$a9=$$α$$9$$−β$$9Now$$, $$considera10$$−$$2a82a9=$$α$$10$$−β$$10$$−$$2$$α$$8$$−β$$82$$α$$9$$−β$$9=$$α$$8$$α$$2$$−$$2$$−β$$8$$β$$2$$−$$22$$α$$9$$−β$$9=$$α$$8$$⋅$$6$$α−β$$8$$⋅$$6$$β$$2$$α$$9$$−β$$9=6$$α$$9$$−$$6$$β$$92$$α$$9$$−β$$9=62=3$$∣∵α and β are the  roots of $$x2$$−$$6x$$−$$2=0$$ or  $$x2=6x+2$$⇒ α$$2=6$$α$$+2$$⇒α$$2$$−$$2=6$$α and β$$2=6$$β$$+2$$⇒β$$2$$−$$2=6$$β

Theory of expressions

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Let $α$ and $β$ be the roots of equation X² - 6x - 2 = O. If $a_{n} = α^{n} - β^{n}, for n \geq 1$ ,then the value $\frac{a_{10} - 2 a_{8}}{2 a_{9}}$ as is equal to

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Theory of expressions

Unlock the full solution & master the concept.

Let α and β be the roots of equation X2 - 6x - 2 = O. If an=αn−βn, for n≥1,then the value a10−2a82a9 as is equal to

Ready to Test Your Skills?

speak to our academic counsellor

free study material

Let $α$ and $β$ be the roots of equation X² - 6x - 2 = O. If $a_{n} = α^{n} - β^{n}, for n \geq 1$ ,then the value $\frac{a_{10} - 2 a_{8}}{2 a_{9}}$ as is equal to