This is a problem from ISI MStat 2018 PSA Problem 12 based on Sequence of positive numbers
Let ,
be a sequence of positive numbers such that
for all n, and
Let
be the polynomial
and suppose
has no real roots for every n . Let
and
be the roots of the polynomial
What can you say about
?
Sequence
Quadratic equation
Discriminant
Answer: is (D)
ISI MStat 2018 PSA Problem 12
Introduction to Real Analysis by Bertle Sherbert
Write the discriminant. Use the properties of the sequence .
Note that as "> has no real root so discriminant is
"> so
"> and
">'s are positive and decreasing so
"> . So , what can we say about a ?
Therefore we can say that hence discriminant of P ,
must be strictly negative so option D.
This is a problem from ISI MStat 2018 PSA Problem 12 based on Sequence of positive numbers
Let ,
be a sequence of positive numbers such that
for all n, and
Let
be the polynomial
and suppose
has no real roots for every n . Let
and
be the roots of the polynomial
What can you say about
?
Sequence
Quadratic equation
Discriminant
Answer: is (D)
ISI MStat 2018 PSA Problem 12
Introduction to Real Analysis by Bertle Sherbert
Write the discriminant. Use the properties of the sequence .
Note that as "> has no real root so discriminant is
"> so
"> and
">'s are positive and decreasing so
"> . So , what can we say about a ?
Therefore we can say that hence discriminant of P ,
must be strictly negative so option D.