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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 \( (\alpha, \beta) \)?

- (A) \( \alpha=\beta, \alpha\) and \(\beta\) are not real
- (B) \( \alpha=\beta, \alpha\) and \(\beta\) are real.
- (C) \(\alpha \neq \beta, \alpha\) and \(\beta\) are real.
- (D) \(\alpha \neq \beta, \alpha\) and \(\beta\) are not real

Sequence

Quadratic equation

Discriminant

But try the problem first...

Answer: is (D)

Source

Suggested Reading

ISI MStat 2018 PSA Problem 12

Introduction to Real Analysis by Bertle Sherbert

First hint

Write the discriminant. Use the properties of the sequence \( a_n \) .

Second Hint

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 ?

Final Step

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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