Master the Geometric Mean of two positive numbers for CBSE Class 11 Applied Mathematics. Learn formulas, properties, and solved examples for your exams.
The Geometric Mean (GM) of two positive numbers a and b is a number G such that the sequence a, G, b forms a Geometric Progression. In such a sequence, the common ratio is constant, implying G/a = b/G. This relationship leads to the fundamental definition where G is the square root of the product of the two numbers.
Step 1: Identify a = 4 and b = 16.
Step 2: Apply the formula G = √ab.
Step 3: G = √(4 × 16) = √64.
Answer: G = 8
The Geometric Mean of two positive numbers always lies between the two numbers themselves. If a < b, then a < √ab < b. Furthermore, the Geometric Mean is always less than or equal to the Arithmetic Mean (AM) of the same two numbers, where AM = (a + b)/2.
Step 1: Calculate GM = √(2 × 8) = √16 = 4.
Step 2: Calculate AM = (2 + 8)/2 = 10/2 = 5.
Step 3: Compare 4 ≤ 5.
Answer: Since 4 < 5, the property holds.
When inserting a single Geometric Mean between two numbers a and b, the resulting sequence is a, G, b. This is a specific case of inserting n geometric means between two numbers. For two numbers, the common ratio r is given by b/G or G/a.
Step 1: Given a = 3, G = 6.
Step 2: Use r = G/a.
Step 3: r = 6/3 = 2.
Answer: The common ratio r is 2.