Let A = n ∈ N ∣ n 2 ≤ n + 10 , 000 , B = 3 k + 1 ∣ k ∈ N and C = 2 k ∣ k ∈ N , then the sum of all the elements of the set A ∩ B - C is equal to ________.

Step-by-step solution

We have, A = n ∈ N ∣ n 2 ≤ n + 10 , 000 , B = 3 k + 1 ∣ k ∈ N and C = 2 k ∣ k ∈ N , B ≡ 4 , 7 , 10 , 13 , 16 , 19 , … … C ≡ 2 , 4 , 6 , 8 , 10 , 12 , 14 , 16 , 20 , … … B - C ≡ 7 , 13 , 19 , … 97 , … Now, n 2 - n ≤ 100 × 100 ⇒ n n - 1 ≤ 100 × 100 ⇒ A = 1 , 2 , … , 100 So, A ∩ B - C = 7 , 13 , 19 , … , 97 Let S n = 7 + 13 + 19 + … + 97 Clearly, the terms are in ari