If arithmetic mean, geometric mean and Harmonic mean between two numbers 'a' and 'b' are A, G and H respectively, then A, G, H will be :

Question

Accepted Answer

To determine the relationship between the arithmetic mean (A), geometric mean (G), and harmonic mean (H) of two numbers 'a' and 'b', we need to understand the definitions and properties of these means:

1. **Arithmetic Mean (A):** The arithmetic mean of two numbers 'a' and 'b' is given by:
   $$
   A = \frac{a + b}{2}
   $$

2. **Geometric Mean (G):** The geometric mean of two numbers 'a' and 'b' is given by:
   $$
   G = \sqrt{ab}
   $$

3. **Harmonic Mean (H):** The harmonic mean of two numbers 'a' and 'b' is given by:
   $$
   H = \frac{2ab}{a + b}
   $$

To determine the series in which A, G, and H are arranged, we need to consider their properties:

- **Geometric Series:** A sequence is in geometric series if each term after the first is the product of the previous term and a constant ratio. For A, G, and H to be in geometric series, the ratio of consecutive terms should be constant.

Let's verify if A, G, and H form a geometric series:

- The ratio of the arithmetic mean to the geometric mean is:
  $$
  \frac{A}{G} = \frac{\frac{a+b}{2}}{\sqrt{ab}} = \frac{a+b}{2\sqrt{ab}}
  $$

- The ratio of the geometric mean to the harmonic mean is:
  $$
  \frac{G}{H} = \frac{\sqrt{ab}}{\frac{2ab}{a+b}} = \frac{(a+b)\sqrt{ab}}{2ab}
  $$

For A, G, and H to be in a geometric series, these two ratios must be equal:
$$
\frac{a+b}{2\sqrt{ab}} = \frac{(a+b)\sqrt{ab}}{2ab}
$$

Simplifying both sides, we find that they are indeed equal, confirming that A, G, and H are in a geometric series.

**Why the other options are incorrect:**

1. **In Harmonic Series:** A harmonic series is characterized by reciprocals of an arithmetic series. A, G, and H do not satisfy this condition.

2. **In Arithmetic Series:** An arithmetic series requires a constant difference between consecutive terms. A, G, and H do not have a constant difference.

3. **Both Harmonic and Geometric Series:** It is not possible for a sequence to simultaneously satisfy the conditions of both harmonic and geometric series, as these are distinct types of sequences.

Therefore, the correct answer is Option 3: A, G, and H are in a geometric series.