Master the Arithmetic Mean of two positive numbers for CBSE Class 11 Applied Mathematics. Learn formulas, properties, and solved practice problems.
For any two positive real numbers a and b, the Arithmetic Mean (AM) is defined as the value that lies exactly midway between them. If A is the AM of a and b, then the sequence a, A, b forms an Arithmetic Progression. This implies that the difference between consecutive terms is constant, specifically A - a = b - A.
Step 1: Identify a = 12 and b = 28.
Step 2: Apply the formula A = (12 + 28) / 2.
Step 3: Calculate the sum 40 and divide by 2.
Answer: A = 20
Inserting an Arithmetic Mean between two numbers a and b creates a three-term Arithmetic Progression. The inserted value A ensures that the common difference d = A - a = b - A. This property is fundamental in extending sequences to include intermediate values while maintaining linear growth.
Step 1: Let a = 5 and b = 15.
Step 2: Calculate A = (5 + 15) / 2 = 10.
Step 3: Verify the sequence 5, 10, 15 has a common difference of 5.
Answer: The AM is 10.
The Arithmetic Mean of two positive numbers is always greater than or equal to the geometric mean of those numbers. Furthermore, the sum of deviations of the numbers from their AM is always zero. This property is essential for understanding the central tendency of a two-term dataset.
Step 1: Find AM: A = (4 + 10) / 2 = 7.
Step 2: Calculate deviations: (4 - 7) = -3 and (10 - 7) = 3.
Step 3: Sum of deviations: -3 + 3 = 0.
Answer: The property holds.