The A.M. of two given positive numbers is 2. If the larger number is increased by 1, the G.M. of the numbers becomes equal to the A.M. of the given numbers. Then find the H.M.

Explanation

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the given information
We know that the Arithmetic Mean (A.M.) of two positive numbers \( a \) and \( b \) is given as 2.

### Step 2: Set up the equation for A.M.
The formula for A.M. of two numbers is:
\[ \text{A.M.} = \frac{a + b}{2} \]
Given that A.M. = 2, we can write:
\[ \frac{a + b}{2} = 2 \]
Multiplying both sides by 2 gives:
\[ a + b = 4 \tag{1} \]

### Step 3: Understand the condition for G.M.
The problem states that if the larger number \( a \) is increased by 1, the Geometric Mean (G.M.) of the numbers becomes equal to the A.M. of the given numbers.

### Step 4: Set up the equation for G.M.
The formula for G.M. of two numbers is:
\[ \text{G.M.} = \sqrt{ab} \]
According to the problem, when \( a \) is increased by 1, we have:
\[ \sqrt{(a + 1)b} = 2 \tag{2} \]

### Step 5: Square both sides of the G.M. equation
Squaring both sides of equation (2) gives:
\[ (a + 1)b = 4 \]
This can be rearranged to:
\[ ab + b = 4 \tag{3} \]

### Step 6: Substitute \( b \) from equation (1) into equation (3)
From equation (1), we have \( b = 4 - a \). Substituting this into equation (3):
\[ a(4 - a) + (4 - a) = 4 \]
Expanding this gives:
\[ 4a - a^2 + 4 - a = 4 \]
Simplifying this leads to:
\[ -a^2 + 3a = 0 \]

### Step 7: Factor the equation
Factoring out \( a \) gives:
\[ a(-a + 3) = 0 \]
This implies:
\[ a = 0 \quad \text{or} \quad a = 3 \]
Since \( a \) must be positive, we take \( a = 3 \).

### Step 8: Find \( b \)
Substituting \( a = 3 \) back into equation (1):
\[ 3 + b = 4 \implies b = 1 \]

### Step 9: Calculate the Harmonic Mean (H.M.)
The formula for H.M. of two numbers \( a \) and \( b \) is:
\[ \text{H.M.} = \frac{2ab}{a + b} \]
Substituting \( a = 3 \) and \( b = 1 \):
\[ \text{H.M.} = \frac{2 \cdot 3 \cdot 1}{3 + 1} = \frac{6}{4} = \frac{3}{2} \]

### Final Answer
The Harmonic Mean (H.M.) of the two numbers is:
\[ \text{H.M.} = \frac{3}{2} \]

---