𝑛2 + 𝑛 = 𝑂(𝑛3). Justify your answer.

The Big O notation is used to describe the upper bound of the time complexity of an algorithm. In this case, we are asked to justify that n^2 + n = O(n^3).

Step 1: Understand the problem We need to prove that the function f(n) = n^2 + n grows no faster than g(n) = n^3 as n approaches infinity.

Step 2: Apply the definition of Big O notation According to the formal definition of Big O notation, f(n) = O(g(n)) if there exist constants c > 0 and n0 such that 0 ≤ f(n) ≤ c*g(n) for all n > n0.

Step 3: Find the constants We need to find constants c and n0 such that 0 ≤ n^2 + n ≤ c*n^3 for all n > n0.

Let's take c = 1 and n0 = 1. For all n > 1, we have:

0 ≤ n^2 + n ≤ n^2 + n^2 = 2n^2 ≤ n^3

Step 4: Conclusion Since we have found constants c and n0 that satisfy the definition of Big O notation, we can conclude that n^2 + n = O(n^3).