A random variable 
following a binomial distribution with success probability 
across 
trials has PMF
where
.
The mean of the distribution is given by the expected value which is defined by
where the summation is carried out over the support of. So the mean is
Because this is a proper distribution, you have
which is a fact that will be used to evaluate the sum above.
Letting
, this becomes
Observe that the remaining sum corresponds to the PMF of a new random variable
which also follows a binomial distribution with success probability , but this time across 
trials. Therefore the sum evaluates to 1, and you're left with 
as the expression for the mean for .
where
The mean of the distribution is given by the expected value which is defined by
where the summation is carried out over the support of
Because this is a proper distribution, you have
which is a fact that will be used to evaluate the sum above.
Letting
Observe that the remaining sum corresponds to the PMF of a new random variable