(Q). If ‘p’ is the number of ways in which three numbers in A.P. can be selected from 1, 2, 3, 4, …….., n. Then find the value of p if
(i). n is odd
(ii). n is even
Solution:-
Given numbers are :-
1, 2, 3, 4, ………., n
Let the three selected numbers in A.P. be a, b, c then
a + c = 2b
Since 2b is even
Hence a+c should also be even
& This is possible only when both a & c are even or both a & c are odd.
Case(i). when n is odd
Let n = 2m + 1
∴ Number of odd numbers in given series =
= m+1
& Number of even numbers in given series =
= m
∴ number of ways of selection of a & c from (m+1) odd integers = ᵐ⁺¹C₂
& number of ways of selection of a & c from m even integers = ᵐC₂
∴ required number of ways = ᵐ⁺¹C₂ × ᵐC₂
=
= m²
= (∵ n = 2m + 1)
= Answer
case(ii). when n is even
we can assume n = 2m
∴ Number of even integers in given series = m
& Number of odd integers in given series = m
∴ Number of ways of selection of a & c from m odd integers = ᵐC₂
Hence total number of ways = ᵐC₂ + ᵐC₂
=
= m(m-1)
= (∵ n = 2m)
= Answer