Question

Prove using the definition of Omega notation that either 8^n is Ω (5^n) or not.

Answer 1

`\(8^n\) is \(\Omega (5^n)\)` according to the definition of Omega notation.

Step-by-step explanation:

The definition of Omega notation requires proving that there exist positive constants
`\(c\) and \(n_0\)` such that
`\(8^n \geq c \cdot 5^n\)` for all
`\(n \geq n_0\).` To demonstrate this, we can simplify the inequality by dividing both sides by
`\(5^n\),` yielding
`\( (8^n)/(5^n) \geq c\).`

Now, let's choose
`\(c = 1\) and \(n_0 = 1\).` For all
`\(n \geq 1\),` the inequality holds:
`\( (8^n)/(5^n) \geq 1\).` This proves that
`\(8^n\) is \(\Omega (5^n)\)` as per the definition.

The choice of
`\(c\) and \(n_0\)` is essential in this proof. In this case, selecting
`\(c = 1\) and \(n_0 = 1\)` simplifies the proof, but it's crucial to note that various choices of constants are possible as long as they satisfy the definition of Omega notation.

The rationale behind the proof lies in demonstrating that the growth rate of
`\(8^n\)` is at least as fast as
`\(5^n\)` multiplied by a positive constant for sufficiently large
`\(n\),` confirming the Omega relationship between the two functions.