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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` If `a_4=28 ,t h e np+2q=` 7 (b) 21 (c) 14 (d) 12

A

14

B

7

C

21

D

12

Text Solution

Verified by Experts

The correct Answer is:
D
`alpha=(a+sqrt5)/(2),beta=(1-sqrt5)/(2)`
`alpha_(4)=alpha_(3)+alpha_(2)`
`=2alpha_(2)+alpha_(1)`
`=3alpha_(1)+2alpha_(0)`
`28=p(3alpha+2)+q(3beta+2)`
`28=(p+q)((3)/(2)+2)+?(p-q)((3sqrt5)/(2))`
`thereforep-q=0`
` and (p+q) xx(7)/(2)=28`
`impliesp+q=8`
`impliesp=q=4`
`thereforep+2q=12`
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