$\begin{array}{1 1}(A)\;\large\frac{5^{k-1}}{4}\\(B)\;5^{k+1}\\(C)\;5^k\\(D)\;\text{None of these}\end{array} $

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The sample space consists of the collection of finite sequences $(x_1,x_2........x_k)$ such that the $k^{th}$ term in each sequence is 2 and all other terms are one of 1,3,4,5,6

$\therefore$ First (k-1) rolls have 5 outcomes each and $k^{th}$ rolls should result in 1 outcomes.

$\therefore$ Event that the 2 appears on the $k^{th}$ roll of the die =$5^{k-1}$ elements.

Similarly for the event that the 2 appears not later than $k^{th}$ roll of the die =$5^0+5^1+5^2+........5^{k-1}$

$\Rightarrow \large\frac{5^{k-1}}{4}$

Hence (A) is the correct answer.

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