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# ISI MStat 2018 PSA Problem 12 | Sequence of positive numbers

This is a problem from ISI MStat 2018 PSA Problem 12 based on Sequence of positive numbers

## Sequence of positive numbers - ISI MStat Year 2018 PSA Question 12

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

### Key Concepts

Sequence

Discriminant

ISI MStat 2018 PSA Problem 12

Introduction to Real Analysis by Bertle Sherbert

## Try with Hints

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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This is a problem from ISI MStat 2018 PSA Problem 12 based on Sequence of positive numbers

## Sequence of positive numbers - ISI MStat Year 2018 PSA Question 12

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

### Key Concepts

Sequence

Discriminant

ISI MStat 2018 PSA Problem 12

Introduction to Real Analysis by Bertle Sherbert

## Try with Hints

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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