1. Show that the following equalities are true:
(a) 5n
2 + 6n = Θ
n
2

(b) n! = O (n
n)
(c) n
2
n
+ 6 · (2n) = Θ
n
2
n 
(d) n
k+ϵ + n
k
log n = Θ
n
k+ϵ

for all fixed k ⩾ 0 and ϵ >

To show that these equalities are true, we will employ the definitions of **Big-O** and **Theta** notation. Here's a step-by-step explanation for each part:

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### (a) \( 5n^2 + 6n = \Theta(n^2) \)

1. **Definition of \( \Theta \)-notation**: \( f(n) = \Theta(g(n)) \) means that there exist constants \( c_1, c_2, n_0 \) such that for all \( n \geq n_0 \):
   \[
   c_1 g(n) \leq f(n) \leq c_2 g(n)
   \]
2. Here, \( f(n) = 5n^2 + 6n \) and \( g(n) = n^2 \).
3. As \( n \) grows large, the term \( 5n^2 \) dominates \( 6n \), meaning that:
   \[
   5n^2 + 6n = \Theta(n^2)
   \]
   - For sufficiently large \( n \), the linear term \( 6n \) becomes negligible compared to \( 5n^2 \), so \( 5n^2 + 6n \) grows asymptotically like \( n^2 \).

Thus, \( 5n^2 + 6n = \Theta(n^2) \).

---

### (b) \( n! = O(n^n) \)

1. **Definition of \( O \)-notation**: \( f(n) = O(g(n)) \) means that there exist constants \( c \) and \( n_0 \) such that for all \( n \geq n_0 \):
   \[
   f(n) \leq c g(n)
   \]
2. Here, \( f(n) = n! \) and \( g(n) = n^n \).
3. Stirling's approximation gives us an approximation for \( n! \):
   \[
   n! \approx \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n
   \]
4. \( \left(\frac{n}{e}\right)^n \) grows slower than \( n^n \), meaning:
   \[
   n! = O(n^n)
   \]
   - This is because \( n! \) grows slower than \( n^n \), and hence we can find a constant \( c \) such that \( n! \leq c \cdot n^n \) for large \( n \).

Thus, \( n! = O(n^n) \).

---

### (c) \( \frac{n^2}{n} + 6 \cdot 2^n = \Theta \left(

Show that the following equalities are true: (a) 5n^2 + 6n = Θ(n^2), (b) n! = O(n^n), (c) n^2/n + 6 · (2^n) = Θ(n^2/n), (d) n^(k+ϵ) + n^k log n = Θ(n^(k+ϵ)) for all fixed k ≥ 0 and ϵ > 0.

Explore how to prove asymptotic equalities such as 5n^2 + 6n = Θ(n^2) and n! = O(n^n) using Big-O and Theta notation. This step-by-step guide covers key mathematical concepts including asymptotic analysis, factorial growth, and polynomial behavior.

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