Combination as Arithmetic Progression

Discrete Mathematics 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 \(n > 2\)?

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