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)`

Explanation

To solve the problem of finding the number of ways to select three numbers in Arithmetic Progression (AP) from the set {1, 2, 3, ..., n} where n is odd, we can follow these steps:

### Step 1: Understand the problem
We need to select three numbers \( a, b, c \) such that they form an arithmetic progression. For three numbers to be in AP, they must satisfy the condition:
\[ b - a = c - b \]
This can be rearranged to:
\[ 2b = a + c \]
This means that \( b \) must be the average of \( a \) and \( c \).
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