If the sum of n arithmetic means between 9 and 51 is 270,

Let a = 9

b = 51

Given sum of n arithmetic means between 9 and 51 is 270.

Let A1, A2, ______ An be arithmetic means between a and b.

∴ a, A1, A2, _____ An , b are in A.P

Here total number of terms is n + 2 first term is a last term is b.

$\therefore$ Sum of first $n+2$ terms $=\frac{(n+2)}{2}\{a+b\}$

$A_{1}+A_{2}+\ldots \ldots \ldots+A_{n}=\frac{n(a+b)}{2}$

i. e $270=n\left(\frac{9+51}{2}\right)$

i. e $270=n\left(\frac{60}{2}\right)$

i. e $\frac{270}{30}=n$

i. e $n=9$

Hence, value of n is 9.