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.

Answer 1

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.