`N (lt 100)` molecules of a gas have velocities 1,2,3….N km/s respectively. Then

Options

A. rms speed and average speed of molecules is same
B. ratio of rms speed to average speed is `sqrt((2N+1))(N+1)//6N`
C. ratio of rms speed to average speed is `sqrt((2N + 1))(N+1)//6`
D. ratio of rms speed to average speed of molecules is `sqrt(((2n+1))/(6(N+1)))`

Explanation

To solve the problem step by step, we need to find the ratio of the root mean square (RMS) speed to the average speed of N molecules of a gas with velocities 1, 2, 3, ..., N km/s.

### Step 1: Calculate the Average Speed
The average speed \( U_{\text{average}} \) of the molecules is given by the formula:
\[ U_{\text{average}} = \frac{U_1 + U_2 + U_3 + \ldots + U_N}{N} \]
Where \( U_i \) are the velocities of the molecules. The sum of the first N natural numbers is:
\[ U_1 + U_2 + \ldots + U_N = 1 + 2 + 3 + \ldots + N = \frac{N(N + 1)}{2} \]
Thus, the average speed becomes:
\[ U_{\text{average}} = \frac{\frac{N(N + 1)}{2}}{N} = \frac{N + 1}{2} \]

### Step 2: Calculate the RMS Speed
The root mean square speed \( U_{\text{RMS}} \) is given by:
\[ U_{\text{RMS}} = \sqrt{\frac{U_1^2 + U_2^2 + U_3^2 + \ldots + U_N^2}{N}} \]
The sum of the squares of the first N natural numbers is:
\[ U_1^2 + U_2^2 + \ldots + U_N^2 = 1^2 + 2^2 + 3^2 + \ldots + N^2 = \frac{N(N + 1)(2N + 1)}{6} \]
Thus, the RMS speed becomes:
\[ U_{\text{RMS}} = \sqrt{\frac{\frac{N(N + 1)(2N + 1)}{6}}{N}} = \sqrt{\frac{(N + 1)(2N + 1)}{6}} \]

### Step 3: Find the Ratio of RMS Speed to Average Speed
Now we need to find the ratio \( \frac{U_{\text{RMS}}}{U_{\text{average}}} \):
\[ \frac{U_{\text{RMS}}}{U_{\text{average}}} = \frac{\sqrt{\frac{(N + 1)(2N + 1)}{6}}}{\frac{N + 1}{2}} \]
This simplifies to:
\[ \frac{U_{\text{RMS}}}{U_{\text{average}}} = \frac{2 \sqrt{\frac{(N + 1)(2N + 1)}{6}}}{N + 1} \]
\[ = \frac{2 \sqrt{(2N + 1)}}{\sqrt{6} \cdot \sqrt{N + 1}} \]

### Step 4: Final Simplification
Thus, the final ratio is:
\[ \frac{U_{\text{RMS}}}{U_{\text{average}}} = \frac{2 \sqrt{2N + 1}}{\sqrt{6(N + 1)}} \]

### Conclusion
The ratio of the RMS speed to the average speed of the molecules is:
\[ \frac{U_{\text{RMS}}}{U_{\text{average}}} = \frac{2 \sqrt{2N + 1}}{\sqrt{6(N + 1)}} \]