This is a problem from ISI MStat 2018 PSA Problem 12 based on Sequence of positive numbers
Let \(a_n \) ,\( n \ge 1\) be a sequence of positive numbers such that \(a_{n+1} \leq a_{n}\) for all n, and \(\lim {n \rightarrow \infty} a{n}=a .\) Let \(p_{n}(x)\) be the polynomial \( p_{n}(x)=x^{2}+a_{n} x+1\) and suppose \(p_{n}(x)\) has no real roots for every n . Let \(\alpha\) and \(\beta\) be the roots of the polynomial \(p(x)=x^{2}+a x+1 .\) What can you say about \( (\alpha, \beta) \)?
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 \( a_n \) .
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 \( 0 \le a < 2 \) hence discriminant of P , \(a^2-4 \) must be strictly negative so option D.
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