Question

If \(a<b\), then \(a<\frac{a+b}{2}<\ldots\)

• A. 2a
• B. 2b
• C. b
• D. None of these

Hint:

Arithmetic mean always lies between two unequal numbers.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a fundamental property of the Arithmetic Mean (AM). The AM of two distinct numbers always lies between the numbers.
Step 2: Detailed Explanation:
Given \( a<b \).
To prove \( a<\frac{a+b}{2} \): Start with \( a<b \). Add \( a \) to both sides: \[ a + a<a + b \implies 2a<a + b \implies a<\frac{a+b}{2} \] To prove \( \frac{a+b}{2}<b \): Start with \( a<b \). Add \( b \) to both sides: \[ a + b<b + b \implies a + b<2b \implies \frac{a+b}{2}<b \] Therefore, \( a<\frac{a+b}{2}<b \).
Step 3: Final Answer:
The expression completes as \( a<\frac{a+b}{2}<b \).