(Q). Find the number of ways in which three numbers in A.P. can be selected from a set of n integers {1, 2, 3, ………..n}.
Solution:-
From the given set let 3 numbers in A.P. be a, b and c.
then b – a = c – b
∴ a + c is even & this can happen by 2 ways
(i) Both of them are even
(ii) Both of them are odd
⦿ when n is even then set has even &
odd numbers.
From this set two even numbers can be selected in ways.
Similarly two odd numbers can be selected in ways.
∴ number of ways when n is even
= +