If `alpha and beta` are the roots of the equation `x^(2)-ax+b=0and A_(n)=alpha^(n)+beta^(n),` then which of the following is true ?

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer

D

Explanation

We have, `alpha + beta = a and alpha beta = b`
`therefore" "alpha^(n+1) + beta^(n+1) = (alpha^(n) + beta^(n))(alpha + beta) - alpha beta(alpha^(n-1) + beta^(n-1))`
`rArr" "V_(n+1) = a V_(n) - b V_(n-1)`
So, statement-2 is true.
Now, `V_(n+1) = a V_(n) - V_(n-1)`
`rArr" "V_(2)=a V_(1) -b V_(0)=a(alpha+beta)-b(alpha^(0)+beta^(0))=a^(2)-2b`
`" "V_(3)=a V_(2) -b V_(1)=a(a^(2)-2b)-ab=a^(3)-3ab`
`" "V_(4)=a V_(3) -b V_(2)=a(a^(3)-3ab)-b(a^(2)-2b)=a^(4)-4a^(2)b+2b^(2)`
and, `V_(5)=a V_(4)-b V_(3) = a(a^(4) -4a^(2)b + 2b^(2))-b(a^(3)-3ab)=a^(5)-5a^(3)b + 5ab^(2)`
So, statement-1 is false.