If p, q > 2 are consecutive in set of primes. Since p,q can only be odd number, (p+q) is an even number.
Can (p+q)/2 be prime?
"There are no primes between p & q..." from @JoeUchill
I think this question is awesome because it reminded me at the first glance that “there are arbitrarily long arithmetic progressions of primes” proven by Green and Tao. http://arxiv.org/abs/math.NT/0404188
So it appeared as a tough question and I immediately gave up...