Question:

Let (xn) (x_n) and (yn) (y_n) be sequences of real numbers defined by
x1=1,y1=12,xn+1=xn+yn2,andyn+1=xnynfor all nN. 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
  • (xn) (x_n) is convergent, but (yn) (y_n) is not convergent.
  • (xn) (x_n) is not convergent, but (yn) (y_n) is convergent.
  • Both (xn) (x_n) and (yn) (y_n) are convergent and limnxn>limnyn\lim_{n \to \infty} x_n > \lim_{n \to \infty} y_n .
  • Both (xn) (x_n) and (yn) (y_n) are convergent and limnxn=limnyn\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 (xn) (x_n) and (yn) (y_n) are convergent and limnxn=limnyn\lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n .
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