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

Options

A. `((n-1)^(2))/(2)`
B. `((n+1)^(2))/(4)`
C. `((n+1)^(2))/(2)`
D. `((n-1)^(2))/(4)`

Answer

D

Explanation

Let n=2m+1. Then, the common difference of the A.P. con be 1, 2, 3,……,m. The number of A.P.'s with 1, 2, 3,…….,m common differences are (2m-1),(2m-3),…….,1 respectiviely.
`:.` Total numbers of A.P.'s `=(2m-1)+......+=m^(2)=((n-1)/(2))^(2)`
`:.` Total number of ways `=((n-1)/(2))^(2)`