How Is the Probability of the First Head on Odd Flips Calculated

[PMH_1]

Can anyone explain to me how this problem is solved

Determine the probability that the first head appears on an odd number of flips i.e. X contains {1,3,5..}.

P[X] = summation starting at x = 1 to infinity (1/2)^(2x-1) = 1/2 / (1 - 1/4) = 2/3

Basically my question is, how is the formula for P[X] obtained:
1. 1/2 / (1 - 1/4) : where does this come from?
2. summation starting at x = 1 to infinity (1/2)^(2x-1) and this as well

 

[mathman]

1. comes from 2. (sum of geometric series).

2. is derived from the fact that you need a sequence of an even number of tails followed by a head. Let n be number of tails P(n=0)=1/2, P(n=2)=1/8, P(n=4)=1/32. In general, P(n=2k)=(1/2)^(2k+1). Sum for k=0, infinity (since events are mutually exclusive and exhaustive). (your x=k+1).