Question

A container is filled with a sample of gas having n molecules with speed a, 2a,3a ....na. the ratio of average speed to root mean square speed is?

Answer 1

Average speed is given as,
V₍avg₎ = a+2a+3a+......... / n
V₍avg₎ = a (1+2+3+........) / n
V₍avg₎ = a / n . n(n+1) / 2
V₍avg₎ = a (n+1) / 2
Now, root mean square speed is given as,
V₍rms₎ = √ (a²+4a²+9a³+.........) / n
V₍rms₎ = √ a² (1+4+9+............) / n
V₍rms₎ = a √ [n (n+1) (2n+1) / 6n]
V₍rms₎ = a √ [(n+1) (2n+1) / 6]
Now,
V₍avg₎ / V₍rms₎ = a(n+1)/2 . 1/a √ [6/(n+1) (2n+1)] 
V₍avg₎ / V₍rms₎ = √[6(n+1)² / 4(n+1)(2n+1)]
V₍avg₎ / V₍rms₎ = √[3(n+1) / 2(2n+1)]
This is the required answer.
Thanks.

Answer 2

for an AP
a = a
d= a
and 
number of terms are n
thus

Sum = n/2 [2a + (n-1)a]
        = n/2 [2a + na - a]
         = n/2 [ a+na]
         = na/2 + n^2a/2