a) Prove that n 3 − 91n 2 − 7n − 14 = Ω(n 3 )....

a) Prove that n 3 − 91n 2 − 7n − 14 = Ω(n 3 ). Your answer must clearly specify the constants c and n0.

b) Let g(n) = 27n 2 + 18n and let f(n) = 0.5n 2 − 100. Find positive constants n0, c1 and c2 such that c1f(n) ≤ g(n) ≤ c2f(n) for all n ≥ n0. Be sure to explain how you arrived at the constants.

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Colby Messinger answered 2 years ago

Prove or disprove each of the followings. If f(n) = ω(g(n)), then log2(f(n)) = ω(log2g(n)), where...

Prove or disprove each of the followings. If f(n) = ω(g(n)), then log2(f(n)) = ω(log2g(n)), where f(n) and g(n) are positive functions. ω(n) + ω(n2) = theta(n). f(n)g(n) = ω(f(n)), where f(n) and g(n) are positive functions. If f(n) = theta(g(n)), then f(n) = theta(20 g(n)), where f(n) and g(n) are positive functions. If there are only finite number of points for which f(n) > g(n), then f(n) = O(g(n)), where f(n) and g(n) are positive functions.

Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 +...

Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 + · · · + n)^2 for every n ∈ N. That is, the sum of the first n perfect cubes is the square of the sum of the first n natural numbers. (As a student, I found it very surprising that the sum of the first n perfect cubes was always a perfect square at all.)

Question

a) Prove that n 3 − 91n 2 − 7n − 14 = Ω(n 3 )....

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