In a through of a die there is 6 sample points.
(i) If 2 appears on the kth roll of the die.
So, first (k-1) roll have 5 outcomes each and kth roll results 2 i.e., 1 outcome.
∴ Number of element of sample space correspond to the event that 2 appears on the kth roll of the die=5k-1
(ii) If we consider that 2 appears later than kth roll of the die. then it is possible that 2 comes in first throw i.e., 1 octcome.
If 2 does not appear in first throw, then outcomes will be 5 and 2 comes in second throw i.i., 1 outcome, possible outcome =5×1=5
Similarly, if 2 does not appear in second throw and appears in third throw.
∴ Possible outcomes=5×5×1
Given Series=1+5+5×5+5×5×5+...+5k-1
=1+5+52+53+...+5k-1
=
1(5k-1)
5-1
=
5k-1
4
