Question 11 Mark
Fill in the blanks:
If P(n)P: 2n < n!, n ∈ N, then P(n) is true for all $\text{n}\geq$ ________.
If P(n)P: 2n < n!, n ∈ N, then P(n) is true for all $\text{n}\geq$ ________.
Answer
View full question & answer→If P(n)P: 2n < n!, n ∈ N, then P(n) is true for all $\text{n}\geq$ 4.
Solution:
Given that P(n): 2n < n!, $\forall\text{ n}\in\text{N}$
For n = 1, 2 < 1 (Not true)
For n = 2, 2 × 2 < 2! ⇒ 4 < 2 (Not true)
For n = 3, 2 × 3 < 3! ⇒ 6 < 3 × 2 × 1 ⇒ 6 < 6 (Not true)
For n = 4, 2 × 4 < 4! ⇒ 8 < 4 × 3 × 2 × 1 ⇒ 8 < 24 (true)
For n = 5, 2 × 5 < 5! ⇒ 10 < 5 × 4 × 3 × 2 × 1 ⇒ 10 < 120 (true)
So, P(n) is true for $\text{n}\geq4$
Hence, the value of the filler is 4.
Solution:
Given that P(n): 2n < n!, $\forall\text{ n}\in\text{N}$
For n = 1, 2 < 1 (Not true)
For n = 2, 2 × 2 < 2! ⇒ 4 < 2 (Not true)
For n = 3, 2 × 3 < 3! ⇒ 6 < 3 × 2 × 1 ⇒ 6 < 6 (Not true)
For n = 4, 2 × 4 < 4! ⇒ 8 < 4 × 3 × 2 × 1 ⇒ 8 < 24 (true)
For n = 5, 2 × 5 < 5! ⇒ 10 < 5 × 4 × 3 × 2 × 1 ⇒ 10 < 120 (true)
So, P(n) is true for $\text{n}\geq4$
Hence, the value of the filler is 4.