The number of ways in which three numbers in A.P. can be selected from 1, 2, 3, …, n is

Given numbers are 1, 2, 3, … n.Let the three selected numbers in A.P. be a, b, c, thenb=a+c2 or a+c=2b----iFrom (1) it is clear that a + c should be an even integer. This is possible only when both a and c are odd or both are even.Case I. When n is even. Let n = 2mThe number of odd numbers = mand number of even numbers = m∴ number of selections of a and c from m odd numbers =  mC2Number of selections of a and c from m even numbers =  mC2∴ Number of ways in this case = 2 ⋅  mC2 = m (m – 1)=n2(n2−1)=n(n−2)4Case II. When n is odd. Let n = 2m + 1Then, number of odd numbers = m + 1and number of even numbers = m∴ Required number in this case =m+1C2+mC2 =(m+1)m2+m(m−1)2=m2=(n−12)2 =14(n−1)2