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?

• A. $(x_n)$ is convergent, but $(y_n)$ is not convergent.
• B. $(x_n)$ is not convergent, but $(y_n)$ is convergent.
• C. Both $(x_n)$ and $(y_n)$ are convergent and $\lim_{n \to \infty} x_n > \lim_{n \to \infty} y_n$.
• D. Both $(x_n)$ and $(y_n)$ are convergent and $\lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n$.

Answer 1

Answer: (D)

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

Answer 2

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