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.

Expert Solution

pls give me positive rating

Colby Messinger answered 2 years ago

Asymptotic Notation. (a) [10 pts.] Prove that n3 −91n2 −7n−14 = Ω(n3). Your answer must clearly...

Asymptotic Notation. (a) [10 pts.] Prove that n3 −91n2 −7n−14 = Ω(n3). Your answer must clearly specify the constants c and n0. (b) [10 pts.] Let g(n) = 27n2 +18n and let f(n) = 0.5n2−100. 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.

1. For each of the following, prove using the definition of O(·): (a) 7n + log(n)...

1. For each of the following, prove using the definition of O(·): (a) 7n + log(n) = O(n) (b) n2 + 4n + 7 = O(n2 ) (c) n! = O(nn) (d) 2n = O(22n) Please explain the procedure clearly for all (They are of the same question)

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 )....

Solutions

Expert Solution

Related Solutions

Asymptotic Notation. (a) [10 pts.] Prove that n3 −91n2 −7n−14 = Ω(n3). Your answer must clearly...

1. For each of the following, prove using the definition of O(·): (a) 7n + log(n)...

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

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

Prove that for every n ∈ N: a) (10^n + 3 * 4^(n+2)) ≡ 4 mod...

series from n=1 to infinity of (4n) / [ (3n^3/2) +7n -9]. the answer should include...

Prove that 3 divides n^3 −n for all n ≥ 1.

Using the definition of Θ notation, prove that (3n + 13)(7n + 2)(log(1024n2 + 100)) ∈...

Prove or disprove that 3|(n 3 − n) for every positive integer n.

Prove by induction that 14^n + 12^n −5^n is divisible by 7 for all n >0