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?
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?
Show Hint
When given sequences defined by the arithmetic and geometric means, the sequences always converge to the same limit.
Updated On: Nov 18, 2025
- Both \( \{a_n\} \) and \( \{b_n\} \) converge, but the limits are not equal
- Both \( \{a_n\} \) and \( \{b_n\} \) converge and the limits are equal
- \( \{b_n\} \) is a decreasing sequence
- \( \{a_n\} \) is an increasing sequence
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The Correct Option is A
Solution and Explanation
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.
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.
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