Let $ (x_n) $ and $ (y_n) $ be sequences of real numbers defined by $$ x_1 = 1, \quad y_1 = \frac{1}{2}, \quad x_{n+1} = \frac{x_n + y_n}{2}, \quad \text{and} \quad y_{n+1} = \sqrt{x_n y_n} \quad \text{for all} \ n \in \mathbb{N}. $$ Then which one of the following is true?

Question

cdquestions Admin · Accepted Answer

The correct option is (D): Both $ (x_n) $ and $ (y_n) $ are convergent and $\lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n $.

Answer

$ (x_n) $ is convergent, but $ (y_n) $ is not convergent.

Answer

$ (x_n) $ is not convergent, but $ (y_n) $ is convergent.

Answer

Both $ (x_n) $ and $ (y_n) $ are convergent and $\lim_{n \to \infty} x_n > \lim_{n \to \infty} y_n $.

Answer

Both $ (x_n) $ and $ (y_n) $ are convergent and $\lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n $.

Let \( (x_n) \) and \( (y_n) \) be sequences of real numbers defined by
\[ x_1 = 1, \quad y_1 = \frac{1}{2}, \quad x_{n+1} = \frac{x_n + y_n}{2}, \quad \text{and} \quad y_{n+1} = \sqrt{x_n y_n} \quad \text{for all} \ n \in \mathbb{N}. \]
Then which one of the following is true?

The Correct Option is D

Solution and Explanation

Top Questions on Sequences and Series

Questions Asked in IIT JAM MA exam

Let \( (x_n) \) and \( (y_n) \) be sequences of real numbers defined by \[ x_1 = 1, \quad y_1 = \frac{1}{2}, \quad x_{n+1} = \frac{x_n + y_n}{2}, \quad \text{and} \quad y_{n+1} = \sqrt{x_n y_n} \quad \text{for all} \ n \in \mathbb{N}. \]Then which one of the following is true?

The Correct Option is D

Solution and Explanation

Top Questions on Sequences and Series

Questions Asked in IIT JAM MA exam

Let \( (x_n) \) and \( (y_n) \) be sequences of real numbers defined by
\[ x_1 = 1, \quad y_1 = \frac{1}{2}, \quad x_{n+1} = \frac{x_n + y_n}{2}, \quad \text{and} \quad y_{n+1} = \sqrt{x_n y_n} \quad \text{for all} \ n \in \mathbb{N}. \]
Then which one of the following is true?