Since $n$ is odd let $n=2m+1$
The common differences of the different $A.P.^s$ can be
$1,2,3,.............m$
The no. of $A.P.^s$ with common differences $1, 2,........m$
respectively are $(2m-1),(2m-3),(2m-5)........1$
$\therefore$ The total no. of $A.P.^s$ =$(2m-1)+(2m-3)+..........+1$
$=m^2=\large\bigg(\frac{n-1}{2}\bigg)^2$