Question

Given below are two statements.

Statement 1: The harmonic mean of \(n\) numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers.

Statement 2: Geometric mean is the arithmetic mean of harmonic mean and arithmetic mean.

In light of the above statements, choose the correct answer from the options given below.

• A. Statement 1 is true but Statement 2 is false.
• B. Statement 1 is false but Statement 2 is true.
• C. Both Statement 1 and Statement 2 are false.
• D. Both Statement 1 and Statement 2 are true.

Hint:

Remember the three important identities for competitive exams:
Arithmetic Mean: \(AM=\dfrac{\sum x_i}{n}\).
Harmonic Mean: reciprocal of the arithmetic mean of reciprocals.
The fundamental relation is \(GM^2=AM\times HM\), not \(GM=\dfrac{AM+HM}{2}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question examines the relationship among the three classical measures of central tendency: Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM).
The first statement is based on the definition of harmonic mean, while the second statement tests the well-known relation connecting AM, GM, and HM.
Key Formula or Approach:
For \(n\) positive observations \(x_1,x_2,\ldots,x_n\):
Arithmetic Mean: \[ AM=\frac{x_1+x_2+\cdots+x_n}{n} \] Harmonic Mean: \[ HM=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}} \] The reciprocal of the arithmetic mean of reciprocals is: \[ \frac{1}{\frac{\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_n}}{n}} =\frac{n}{\sum \frac1{x_i}} =HM \] Also, the correct relationship among the three means is: \[ GM^2=AM\times HM \]

Step 2: Detailed Explanation:
Statement 1:
The harmonic mean is defined as the reciprocal of the arithmetic mean of the reciprocals of the given numbers.
From the formula, \[ HM=\frac{n}{\sum \frac1{x_i}} =\frac{1}{\frac1n\sum\frac1{x_i}} \] Hence, Statement 1 exactly matches the mathematical definition.
Therefore, Statement 1 is true.
Statement 2:
The statement claims that the geometric mean is the arithmetic mean of HM and AM, i.e., \[ GM=\frac{AM+HM}{2} \] This is incorrect.
The actual relation is: \[ GM=\sqrt{AM\times HM} \] For example, let the two numbers be \(2\) and \(8\).
Their arithmetic mean is: \[ AM=\frac{2+8}{2}=5 \] Their harmonic mean is: \[ HM=\frac{2\times2\times8}{2+8} =\frac{32}{10}=3.2 \] Their geometric mean is: \[ GM=\sqrt{2\times8} =\sqrt{16}=4 \] Now, \[ \frac{AM+HM}{2} =\frac{5+3.2}{2} =4.1 \] Since \(4\neq4.1\), the statement is false.
Therefore, Statement 2 is false.

Step 3: Final Answer:
Statement 1 is true, whereas Statement 2 is false.
Hence, the correct option is: \[ \boxed{(A) Statement 1 is true but Statement 2 is false. \]