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Let p ,q be integers and let alpha,beta ...

Let `p ,q` be integers and let `alpha,beta` be the roots of the equation, `x^2-x-1=0,` where `alpha!=beta` . For `n=0,1,2, ,l e ta_n=palpha^n+qbeta^ndot` FACT : If `aa n db` are rational number and `a+bsqrt(5)=0,t h e na=0=bdot`

A

`a_(11)+2alpha_(10)`

B

`2a_(11)+2a_(10)`

C

`a_(11)-a_(10)`

D

`a_(11)+a_(10)`

Text Solution

Verified by Experts

The correct Answer is:
D
`alpha^(2)=alpha+1`
`beta^(2)=beta+1`
`a_(n)=palpha^(n)=qbeta^(n)`
`=p(alpha^(n-1)+alpha^(n-2))+q(beta^(n-1)+beta^(n-2))`
`=a_(n-1)+a_(n-2)`
`thereforea_(12)=a_(11)+a_(10)`
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