Let N be the sum of the numbers appeared when two fair dice are...

Given,

Two fair dices are rolled,

So total sample space will be $n\left(s\right) = 36$

Now given: $N - 2, \sqrt{3N}, N + 2$ are in G.P.

So, $3N = \left(N - 2\right)\left(N + 2\right)$

$⇒ 3N = {N}^{2} - 4$

$⇒ {N}^{2} - 3N - 4 = 0$

$⇒ \left(N - 4\right)\left(N + 1\right) = 0$

$⇒ N = 4$ or $N = - 1$ (rejected)

Now favourable cases for sum $4 ≡ \left\{\left(1, 3\right), \left(3, 1\right), \left(2, 2\right)\right\}$

So, $n\left(A\right) = 3$

Hence, $P\left(A\right) = \dfrac{3}{36} = \dfrac{1}{12} = \dfrac{4}{48} ⇒ k = 4$