Combination as Arithmetic Progression

Probability Level 5

What is the least value of p p such that (pq),(pq+1),(pq+2)\dbinom{p}{q}, \dbinom{p}{q + 1}, \dbinom{p}{q + 2} is an arithmetic progression for some positive integers pp and qq satisfying 1qp2? 1 \leq q \leq p - 2 ?

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

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

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