Question:
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?
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?
\[ x_1 = 1, \quad y_1 = \frac{1}{2}, \quad x_{n+1} = \frac{x_n + y_n}{2}, \quad \text{and} \quad y_{n+1} = \sqrt{x_n y_n} \quad \text{for all} \ n \in \mathbb{N}. \]
Then which one of the following is true?
Updated On: Dec 24, 2024
- \( (x_n) \) is convergent, but \( (y_n) \) is not convergent.
- \( (x_n) \) is not convergent, but \( (y_n) \) is convergent.
- Both \( (x_n) \) and \( (y_n) \) are convergent and \(\lim_{n \to \infty} x_n > \lim_{n \to \infty} y_n \).
- Both \( (x_n) \) and \( (y_n) \) are convergent and \(\lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n \).
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The Correct Option is D
Solution and Explanation
The correct option is (D): Both \( (x_n) \) and \( (y_n) \) are convergent and \(\lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n \).
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