Combination as Arithmetic Progression

Probability Level 5

What is the least value of $p$ such that $\dbinom{p}{q}, \dbinom{p}{q + 1}, \dbinom{p}{q + 2}$ is an arithmetic progression for some positive integers $p$ and $q$ satisfying $1 \leq q \leq p - 2 ?$

Notation: $\binom MN = \frac {M!}{N! (M-N)!}$ denotes the binomial coefficient.

Bonus: Is it possible for the following progression $\dbinom{p}{q}, \dbinom{p}{q + 1}, \dbinom{p}{q + 2},\ldots,\dbinom{p}{q+n}$ to exist for $q + n \leq p$ and $n > 2$ ?