1) Consider the assertions below. Prove or disprove the assertion using limits, possibly with L’Hoˆpital’s rule....

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1) Consider the assertions below. Prove or disprove the assertion using limits, possibly with L’Hoˆpital’s rule....

1) Consider the assertions below. Prove or disprove the assertion using limits, possibly with L’Hoˆpital’s rule. Also, if the assertion is true, show that it is true directly from the definition of the asymptotic notation and derive values for the relevant constants.

(a) 3n^2 + 5n + 7 ∈ O(n^2)

(b) 5(n − 2)! ∈ Θ(n!)

(c) $^3\sqrt[\sqrt[]{n^3-5}{}$ ∈ Θ(n)

2) Give the recurrence relation where indicated or solve the given recurrence relation by algebraically unrolling it.

(a) Give a recurrence relation, T(n), and its initial condition for the sequence 3, 5, 9, 17, 33, 65 ··· .

(b) Solve the relation in (a).

(c) Solve T(n) = cn + T(n − 1) with the initial condition T(1) = 1.

(d) Solve T(n) = αT(n / α) + βn with the initial condition T(1) = 1.

engineering Computer-Science

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eim 3 isa finite Hent value Here n-2 im f(n Hence

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