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

Options

A. 21
B. 14
C. 7
D. 12

Answer

4

Explanation

`a_(4) = a_(3) + a_(2)`
` = a_(2) + a_(1) + a_(1) + a_(0)`
`a_(1) + a_(0) + 2a_(1) + a_(0)`
` 2a_(0) + 3a_(1) = 2(p+ q) + 3 (p alpha + q beta)`
` = 2(p + q) + 3 (p alpha + q) (1 - alpha))`
= ` (2p + 5q = 28 and 3p - 3q = 0 as p,q in Q and alpha ` is irrational .
` therefore p = q = 4 `
`therefore =P + 2q = 12 `