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Let alpha and beta be the roots of equat...

Let `alpha and beta` be the roots of equation `x^(2)-6x-2=0. If a_(n)=alpha^(n)-beta^(n),"for"nge1,` then the value of `(a_(10)-2a_(8))/(2a_(9))` is

A

1

B

2

C

3

D

4

Text Solution

Verified by Experts

The correct Answer is:
C
`(a_(10)-2a_(8))/(2a_(8))=((alpha^(10)-beta^(10))-2(alpha^(8)-beta^(8)))/(2(alpha^(9)-beta^(9)))`
`=(alpha^(8)(alpha^(2)-2)-beta^(8)beta^(2)-2)/(2(alpha^(9)-beta^(9)))`
`because alpha"is root of"x^(2)-6x2=0impliesalpha^(2)-2=6alpha[andbeta"is root of"x^(2)-6x-2=0impliesbeta^(2)-2=6beta]`
`=(alpha^(8)(6 alpha)-beta^(8)(6beta))/(2(alpha^(9)-beta^(9)))=(6(alpha^(9)-beta^(9)))/(2(alpha^(9)-beta^(9)))=3`
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