If a = the Nth prime number and b = the (n/2+1)th prime number, will b always be larger than a/2?

I'm learning to program and wrote a script to find prime numbers that works perfectly only if this is true. I've tested up to the the 17666th prime and it seems to work, but it doesn't feel right to just assume.

Also can something similar be said of a/3, a/5, a/7, a/11 etc...