KVPY2019MathematicsSets and Relations
Let N be the set of positive integers. For all n ∈ N , let f n = n + 1 1 / 3 - n 1 / 3 and A = n ∈ N : f n + 1 < 1 3 n + 1 2 / 3 < f n Then,
Options
- AA = N
- BA is a finite set
- Cthe complement of A in N is nonempty, but finite
- DA and its complement in N are both infinite
Correct answer
A. A = N
Step-by-step solution
It is given that for n ∈ N f n = n + 1 1 / 3 − n 1 / 3 = n + 1 − n n + 1 2 / 3 + n + 1 2 / 3 n 2 / 3 + n 2 / 3 = 1 n + 1 2 / 3 + n + 1 2 / 3 n 2 / 3 + n 2 / 3 ∵     ∀ n ∈ N 3 n 2 / 3 < n + 1 2 / 3 + n + 1 2 / 3 n 2 / 3 + n 2 / 3 < 3 n + 1 2 / 3 ⇒ 1 3 n + 1 2 / 3 < 1 + n + 1 2 / 3 + n + 1 2 / 3 n 2 / 3 + n 2 / 3 < 1 3 n 2 / 3 ⇒ 1 3 n + 1 2 / 3 < f n < 1 3 n 2 / 3 Similarly, 1 3 n + 2 2 / 3 < f n + 1 < 1 3 n + 1 2 / 3 ∴ &#