$$N\left( { < 100} \right)$$   molecules of a gas have velocities $$1,2,3,\,.........N\,km/s$$     respectively. Then ratio of $$rms$$  speed and average speed is

Solution :
$$\eqalign{ & {V_{rms}} = \sqrt {\frac{{{1^2} + {2^2} + ....... + {N^2}}}{{\;N}}} \cr & = \sqrt {\frac{{N\left( {N + 1} \right)\left( {2\;N + 1} \right)}}{{6\;N}}} \cr & {V_{{\text{mean}}}} = \frac{{1 + 2 + ......N}}{N} = \frac{{N\left( {N + 1} \right)}}{{2\;N}} \cr} $$