Question

Select the correct inequality for the asymptotic order of growth of the function n²ⁿ – 2ⁿ?

1) n²ⁿ = O(2ⁿ)
2) n²ⁿ = Ω(2ⁿ)
3) n²ⁿ = Θ(2ⁿ)
4) n²ⁿ = o(2ⁿ)

Answer 1

The correct inequality for the asymptotic order of growth of the function n²ⁿ - 2ⁿ is n²ⁿ = Ω(2ⁿ), indicating that n²ⁿ grows at least as fast as 2ⁿ for large values of n. The correct answer is 2.

Step-by-step explanation:

The correct inequality for the asymptotic order of growth of the function n²ⁿ - 2ⁿ is n²ⁿ = Ω(2ⁿ), indicating that n²ⁿ grows at least as fast as 2ⁿ for large values of n.

The student has posed a question regarding the asymptotic order of growth of the function n²ⁿ - 2ⁿ. To ascertain the correct inequality that describes the relationship between n²ⁿ and 2ⁿ, we must delve into Big O notation, which is widely utilized in computer science to describe the upper bounds of algorithm complexities.

In comparing n²ⁿ and 2ⁿ, it is evident that as n tends towards infinity, the term n²ⁿ grows much faster than 2ⁿ due to its additional multiplicative factor of n². Thus, the correct inequality to describe this relationship is n²ⁿ = Ω(2ⁿ), which asserts that n²ⁿ grows at least as fast as 2ⁿ when n is significantly large.