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JEE Advance - Mathematics (2017 - Paper 2 Offline - No. 17)

Let p, q be integers and let $$\alpha $$, $$\beta $$ be the roots of the equation, x2 $$-$$ x $$-$$ 1 = 0 where $$\alpha $$ $$ \ne $$ $$\beta $$. For n = 0, 1, 2, ........, let an = p$$\alpha $$n + q$$\beta $$n.

FACT : If a and b are rational numbers and a + b$$\sqrt 5 $$ = 0, then a = 0 = b.
Let p, q be integers and let $$\alpha $$, $$\beta $$ be the roots of the equation, x2 $$-$$ x $$-$$ 1 = 0 where $$\alpha $$ $$ \ne $$ $$\beta $$. For n = 0, 1, 2, ........, let an = p$$\alpha $$n + q$$\beta $$n.

FACT : If a and b are rational numbers and a + b$$\sqrt 5 $$ = 0, then a = 0 = b.
Let p, q be integers and let $$\alpha $$, $$\beta $$ be the roots of the equation, x2 $$-$$ x $$-$$ 1 = 0 where $$\alpha $$ $$ \ne $$ $$\beta $$. For n = 0, 1, 2, ........, let an = p$$\alpha $$n + q$$\beta $$n.

FACT : If a and b are rational numbers and a + b$$\sqrt 5 $$ = 0, then a = 0 = b.
a12 = ?
a11 + 2a10
2a11 + a10
a11 $$-$$ a10
a11 + a10

Explanation

$$\alpha $$2 = $$\alpha $$ + 1

$$\beta $$2 = $$\beta $$ + 1

an = p$$\alpha $$n + q$$\beta $$n

= p($$\alpha $$n$$-$$1 + $$\alpha $$n$$-$$2) + q($$\beta $$n$$-$$1 + $$\beta $$n$$-$$2)

= an$$-$$1 + an$$-$$2

$$ \therefore $$ a12 = a11 + a10

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