If `A_(1),A_(2)` are two A.M.'s and `G_(1),G_(2)` be two G.M.'s between two positive numbers a and b, then `(A_(1)+A_(2))/(G_(1)G_(2))` is equal to
(i) `(a+b)/(ab)`
(ii) `(a+b)/2`
(iii) `(a+)/(a-b)`
(iv) None of these

Options

A. `(a+b)/(ab)`
B. `(a+b)/2`
C. `(a+)/(a-b)`
D. None of these

Answer

To solve the problem step by step, we need to find the expression \((A_1 + A_2) / (G_1 G_2)\) given that \(A_1\) and \(A_2\) are two arithmetic means (A.M.) and \(G_1\) and \(G_2\) are two geometric means (G.M.) between two positive numbers \(a\) and \(b\).

### Step 1: Define the Arithmetic Means
The arithmetic means \(A_1\) and \(A_2\) can be defined as follows:
- Let the first number be \(a\), the second number be \(b\), and the two arithmetic means be \(A_1\) and \(A_2\).
- The common difference \(d\) is given by:
\[ d = \frac{b - a}{3} \]
- Therefore, we can express \(A_1\) and \(A_2\) as:
\[ A_1 = a + d = a + \frac{b - a}{3} = \frac{3a + b - a}{3} = \frac{2a + b}{3} \]
\[ A_2 = a + 2d = a + 2 \cdot \frac{b - a}{3} = a + \frac{2(b - a)}{3} = \frac{3a + 2b - 2a}{3} = \frac{a + 2b}{3} \]

### Step 2: Calculate \(A_1 + A_2\)
Now, we can find \(A_1 + A_2\):
\[ A_1 + A_2 = \frac{2a + b}{3} + \frac{a + 2b}{3} = \frac{(2a + b) + (a + 2b)}{3} = \frac{3a + 3b}{3} = a + b \]

### Step 3: Define the Geometric Means
Next, we define the geometric means \(G_1\) and \(G_2\):
- The common ratio \(r\) is given by:
\[ r = \sqrt[3]{\frac{b}{a}} \]
- Thus, we can express \(G_1\) and \(G_2\) as:
\[ G_1 = a \cdot r = a \cdot \left(\frac{b}{a}\right)^{1/3} = a^{2/3} b^{1/3} \]
\[ G_2 = a \cdot r^2 = a \cdot \left(\frac{b}{a}\right)^{2/3} = a^{1/3} b^{2/3} \]

### Step 4: Calculate \(G_1 G_2\)
Now we find \(G_1 G_2\):
\[ G_1 G_2 = (a^{2/3} b^{1/3})(a^{1/3} b^{2/3}) = a^{(2/3 + 1/3)} b^{(1/3 + 2/3)} = a^{1} b^{1} = ab \]

### Step 5: Calculate \((A_1 + A_2) / (G_1 G_2)\)
Now we can substitute \(A_1 + A_2\) and \(G_1 G_2\) into the expression:
\[ \frac{A_1 + A_2}{G_1 G_2} = \frac{a + b}{ab} \]

### Conclusion
Thus, the final answer is:
\[ \frac{A_1 + A_2}{G_1 G_2} = \frac{a + b}{ab} \]

### Answer
The answer is option (i) \(\frac{a + b}{ab}\).