Given that n is odd, the number of ,ways in which numbers in AP can be selected from 1, 2, 3, ... , n, is

Let $n=2m+1$. Then, the common difference of the  A.P. can be 1,2,3,…m. The number of A.P.'s with 1, 2, 3, ... , m common differences are (2m -1), (2m - 3), ... , 1 respectively

Total number of A.P.'s $=(2m-1)+\dots .+1=m^2={\left(\frac{n-1}{2}\right)}^2$

Total number of ways $={\left(\frac{n-1}{2}\right)}^2$