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Question
Give an example of a statement P(n) which is true for all n ≥ 4 but P(1), P(2) and P(3) are not true. Justify your answer.
Solution
Let P(n) be the statement 3n < n!.
For n = 1,
3n = 3 × 1 = 3
n! = 1! = 1
Now, 3 > 1
So, P(1) is not true.
For n = 2,
3n = 3 × 2 = 6
n! = 2! = 2
Now, 6 > 2
So, P(2) is not true.
For n = 3,
3n = 3 × 3 = 9
n! = 3! = 6
Now, 9 > 6
So, P(3) is not true.
For n = 4,
3n = 3 × 4 = 12
n! = 4! = 24
Now, 12 < 24
So, P(4) is true.
For n = 5,
3n = 3 × 5 = 15
n! = 5! = 120
Now, 15 < 120
So, P(5) is true.
Similarly, it can be verified that 3n < n! for n = 6, 7, 8, ... .
Thus, the statement P(n) : 3n < n! is true for all n ≥ 4 but P(1), P(2) and P(3) are not true.
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