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

The Correct Option is D

To determine which statement about the sequences \((x_n)\) and \((y_n)\) is true, let's analyze their behavior using the given recursive formulae:

1. Initially, we have:
   • \(x_1 = 1\)
   • \(y_1 = \frac{1}{2}\)
2. The recursive definitions are:
   • \(x_{n+1} = \frac{x_n + y_n}{2}\)
   • \(y_{n+1} = \sqrt{x_n y_n}\)
3. Next, we verify if both sequences are bounded and monotonic:
   • Clearly, \(x_n \geq y_n\) for all \(n\) as \(x_1 = 1 > \frac{1}{2} = y_1\), and both updates ensure \(x_{n+1} \geq y_{n+1}\).
   • The sequence \((x_n)\) is bounded below by \(y_1 = \frac{1}{2}\) and above by \(x_1 = 1\). It is monotonic because: \(x_{n+1} = \frac{x_n + y_n}{2} \leq x_n\) if \(x_n \geq y_n\).
   • The sequence \((y_n)\) is increasing and bounded above because: \(y_{n+1} = \sqrt{x_n y_n} \geq y_n\).
4. Since \((x_n)\) is decreasing and bounded, it converges by the Monotone Convergence Theorem.
5. Since \((y_n)\) is increasing and bounded, it also converges by the Monotone Convergence Theorem.
6. For convergence of both:
   • Let \(L = \lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n\).
   • At convergence:
     • \(L = \frac{L + L}{2} = L\)
     • \(L = \sqrt{L \cdot L} = L\)

Thus, both sequences \((x_n)\) and \((y_n)\) are convergent and \(\lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n\). Therefore, the correct option is:

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