Question

Permutations and combinations.
How many Arithmetic Progressions are there in an AP of n terms.

Answer 1

@KingGeorge Help?

Answer 2

probably fo consecutive terms
\[ \sum_{i=0}^n \binom{n-(3+i)}{1} \]

Answer 3

with spacing 2
\[ 2 \times \sum_{i=0}^{\left \lfloor n \over 2 \right \rfloor} \binom{\left \lfloor n \over 2 \right \rfloor-(3+i)}{1} \]

with spacing 3
\[ 3 \times \sum_{i=0}^{\left \lfloor n \over 3 \right \rfloor} \binom{\left \lfloor n \over 3 \right \rfloor-(3+i)}{1} \]

Answer 4

looks like i should consider cases ...

Answer 5

the general case seems difficult .. anyway this serves as bookmark.