Question

Using the definition of Big-Oh discussed in class, to prove that 10n = O(n²) we can select ___.

(a)c = 0, n₀= 1

(b)c = 1, n₀= 1

(c)c = 2, n₀= 5

(d)c = 1, n₀= 9

Would really appreciate an explanation along with the correct answer, as I am trying to learn. Thank you.

Answer 1

f(n) = O(g(n)) means there are positive constants c and n0, such that 0 ≤ f(n) ≤ cg(n) for all n ≥ n0

10n = O(n^2)
=>  10n <= c(n^2)
Let's assume c = 2
=>  10n <= c(n^2)
=>  10n <= 2(n^2)
=>  5 <= n

it's true for all n >= 5

so, 10n = O(n^2) given c = 2, n0 = 5

Answer: (c).    c = 2, n0= 5