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Question

Let p,q be integers and let α,β be the roots of the equation, x2x1=0, where αβ. For n=0,1,2,...., let an=pαn+qβn.

FACT: If a and b are rational numbers and a+b5=0. then a=0=b.

a12=

A
a11a10
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B
a11+a10
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C
2a11+a10
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D
a11+2a10
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Solution

The correct option is B a11+a10
α,β are the roots of the equation x2x1=0
Therefore, α2α1=0α2=α+1
and β2β1=0β2=β+1

an+1+an=pαn+1+qβn+1+pαn+qβnan+1+an=pαn(α+1)+qβn(β+1) =pαnα2+qβnβ2=pαn+2+qβn+2
an+1+an=an+2
a11+a10=a12

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