Question

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.

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
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 ≤...
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 ≤...
Let N2K be the set of the first 2k natural numbers. Prove that if we choose...
Let N2K be the set of the first 2k natural numbers. Prove that if we choose k + 1 numbers out of these 2k, there is at least one pair of numbers a, b for which a is divisible by b.
For n in natural number, let A_n be the subset of all those real numbers that...
For n in natural number, let A_n be the subset of all those real numbers that are roots of some polynomial of degree n with rational coefficients. Prove: for every n in natural number, A_n is countable.
Prove the following using induction: (a) For all natural numbers n>2, 2n>2n+1 (b) For all positive...
Prove the following using induction: (a) For all natural numbers n>2, 2n>2n+1 (b) For all positive integersn, 1^3+3^3+5^3+···+(2^n−1)^3=n^2(2n^2−1) (c) For all positive natural numbers n,5/4·8^n+3^(3n−1) is divisible by 19
Let S be the set {(-1)^n +1 - (1/n): all n are natural numbers}. 1. find...
Let S be the set {(-1)^n +1 - (1/n): all n are natural numbers}. 1. find the infimum and the supremum of S, and prove that these are indeed the infimum and supremum. 2. find all the boundary points of the set S. Prove that each of these numbers is a boundary point. 3. Is the set S closed? Compact? give reasons. 4. Complete the sentence: Any nonempty compact set has a....
Exercise 6.6. Let the inductive set be equal to all natural numbers, N. Prove the following...
Exercise 6.6. Let the inductive set be equal to all natural numbers, N. Prove the following propositions. (a) ∀n, 2n ≥ 1 + n. (b) ∀n, 4n − 1 is divisible by 3. (c) ∀n, 3n ≥ 1 + 2 n. (d) ∀n, 21 + 2 2 + ⋯ + 2 n = 2 n+1 − 2.
Let S(n) be the statement: The sum of the first n natural numbers is 1/2 n2...
Let S(n) be the statement: The sum of the first n natural numbers is 1/2 n2 + 1/2 n + 1000. Show that if S(k) is true, so is S(k+1).
Let f(n) be a negligible function and k a positive integer. Prove the following: (a) f(√n)...
Let f(n) be a negligible function and k a positive integer. Prove the following: (a) f(√n) is negligible. (b) f(n/k) is negligible. (c) f(n^(1/k)) is negligible.
Let (a, b, c, d) be elements within N+ (all positive natural numbers). If a is...
Let (a, b, c, d) be elements within N+ (all positive natural numbers). If a is less than/equal to b, and c is less than/equal to d, then (a*c) is less than/equal to (b*d) with equality if and only if a=b and c=d. Proof?