Given two numbers a and b. Let A denote the single A.M. and S denote the sum of n A.M.'s between a and b, then `S//A` depends on

Options

A. n,a,b
B. n,b
C. n,a
D. n

Answer

To solve the problem, we need to analyze the relationship between the arithmetic mean (A.M.) of two numbers \( a \) and \( b \), and the sum \( S \) of \( n \) arithmetic means between them.

### Step-by-Step Solution:

1. **Identify the Single A.M. (A)**:
The single arithmetic mean \( A \) of the two numbers \( a \) and \( b \) is given by:
\[ A = \frac{a + b}{2} \]

2. **Define the n A.M.s**:
Let the \( n \) arithmetic means between \( a \) and \( b \) be denoted as \( a_1, a_2, \ldots, a_n \). These means form an arithmetic progression (A.P.) with the first term \( a_1 = a \) and the last term \( a_n = b \).

3. **Sum of n A.M.s (S)**:
The sum \( S \) of the \( n \) arithmetic means can be calculated using the formula for the sum of an arithmetic series:
\[ S = \frac{n}{2} \times (\text{first term} + \text{last term}) = \frac{n}{2} \times (a + b) \]

4. **Calculate \( S/A \)**:
Now, we need to find the ratio \( \frac{S}{A} \):
\[ \frac{S}{A} = \frac{\frac{n}{2} \times (a + b)}{\frac{a + b}{2}} \]

5. **Simplifying the Expression**:
When we simplify the above expression, we can cancel \( \frac{a + b}{2} \) in the numerator and denominator:
\[ \frac{S}{A} = \frac{n}{2} \times \frac{2}{1} = n \]

6. **Conclusion**:
Thus, we find that:
\[ \frac{S}{A} = n \]
This indicates that \( S/A \) depends solely on \( n \).

### Final Answer:
The value of \( \frac{S}{A} \) depends on \( n \) alone.

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