AP EAMCET202119 Aug 2021Evening ShiftMathematicsMathematical InductionActual
For what natural numbers n ∈ N , the inequality 2 n > n + 1 is valid?
Options
- A∀ n ∈ N
- B∀ n ≥ 2
- C∀ 1 ≤ n ≤ 3
- D∀ n ∈ N − 2 , 3
Correct answer
B. ∀ n ≥ 2
Step-by-step solution
Given that, 2 n > n + 1 For n = 1 , 2 1 > 1 + 1 ⇒ 2 > 2 is not true For n = 2 , 2 2 > 2 + 1 ⇒ 4 > 3 is True. So, Given inequality is true for n = 2 . Assume that 2 m > m + 1 is true for m > 2 . But 2 m > 1 for all m > 1 ⇒ 2 m + 2 m > m + 1 + 1 ⇒ 2 m + 1 > m + 1 + 1 Hence, the inequality holds for n = m + 1 It follows by mathematical induction that the given inequality holds for all natural numbers n ≥ 2 .