Question

Let \( 0<a_1<b_1 \), For \( n \geq 1 \), define \[ a_{n+1} = \sqrt{a_n b_n} \quad \text{and} \quad b_{n+1} = \frac{a_n + b_n}{2}. \] Then which one of the following is NOT TRUE?

• A. Both \( \{a_n\} \) and \( \{b_n\} \) converge, but the limits are not equal
• B. Both \( \{a_n\} \) and \( \{b_n\} \) converge and the limits are equal
• C. \( \{b_n\} \) is a decreasing sequence
• D. \( \{a_n\} \) is an increasing sequence

Hint:

When given sequences defined by the arithmetic and geometric means, the sequences always converge to the same limit.

Answer 1

The Correct Option is A

Step 1: Understand the sequences.
The sequences \( \{a_n\} \) and \( \{b_n\} \) are defined recursively. The sequence \( \{a_n\} \) is defined as the geometric mean of \( a_n \) and \( b_n \), and \( \{b_n\} \) is defined as the arithmetic mean of \( a_n \) and \( b_n \).
Step 2: Analyze the behavior of the sequences.
It is known that both sequences \( \{a_n\} \) and \( \{b_n\} \) converge to the same limit. This result follows from the fact that the arithmetic mean is greater than or equal to the geometric mean, and the sequences are bounded and monotonic.
Step 3: Conclusion.
Thus, the correct answer is (A), because the limits of both sequences are actually equal, contrary to what option (A) claims.