Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.
In y = ax +b when are a, -b/a, b in arithmetic progression. The polynomial y = ax^2 + bx + c has roots r1 and r2. Can a, r1, b, r2 and c be in arithmetic progression?
What can you say about the common difference of an AP where every term is prime?
Investigate special cases for small $n$ first.
Can you spot patterns in the sums for $n=3$, $4$, $5$, $6$, $7 \dots$?