I'm studying for my exam of probability distributions and in my study book got these equalities:
Also mention that: is equal to
how to reach these inequalities?
There appears to be a typo in the equations (not "equalities"). As Dilip pointed out in the comments, they are false as written. However, they become rather less mysterious if the lower limit of the sum is $i$ instead of $1$, which indeed seems to be the intention, judging from your comments. (It would have been preferable if you'd corrected the equation when Dilip pointed out the error rather than just responding in a comment.)
With this change, this is merely an expression for the average of the numbers from $i$ to $j$, since there are $i-j+1$ of these. Clearly the average of the numbers from $i$ to $j$ is halfway between them at $(i+j)/2$.