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Let S(n) be a monotonic non-decreasing positive function defined for all natural numbers n. We do...

Let S(n) be a monotonic non-decreasing positive function defined for all natural numbers n. We do not know the value of S(n) for every n ∈ N except when n = 2k for some k ∈ N, in which case S(n) = n log n + 3n − 5. Show that S(n) ∈ Θ(n log n).

Hint: (if you use it, you need to prove it): ∀n > 1 ∈ N, ∃k ∈ N, such that 2k-1 ≤ n ≤ 2k.

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