Answer:
For all n ≥ 4, 2n < n!
Step-by-step explanation:
Let's use the induction method to prove this statement.
In the induction method, first we prove the statement for n=4
1) If n = 4 ⇒2(4) < 4! ⇒2(4) < 24 ⇒8 < 24.
Therefore the statement holds for n=4
2) Now we assume that the statement is valid for n = k
⇒2k < k!
3) Now we will prove the statement holds for n = k +1
We will prove that 2(k + 1) < (k +1)!
(k + 1)! = (k+1) (k) (k-1) .... (3) (2) (1)
If the statement is valid for k + 1, then it would mean that
2 (k + 1) < (k+1) (k) (k-1) ... (3) (2) (1)
2 < (k) (k-1).... (3) (2) (1)
which is clearly true since k ≥4
Therefore the statement n ≥4, 2n < n! is true.
