Problem:A positive integer $k$ greater than $1$ is given. Prove that there exist a prime $p$ and a strictly increasing sequence of positive integers $a_1, a_2, . . . , a_n, . . .$ such that the terms of the sequence $p + ka_1, p + ka_2, . . . , p + ka_n, . . .$ are all primes.
I have never encountered problems of this kind before and therefore I don't even know how to start. I am guessing that we need to provide some sort of construction for $a_n$, but I'm not sure how to proceed further.