Give an example of a statement P(n) which is true for all n ≥ 4 but

1.	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» According to the question, P(n) which is true for all n ≥ 4 but P(1), P(2) and P(3) are not true Let P(n) be 2ⁿ < n! So, the examples of the given statements are, P(0) ⇒ 2⁰ < 0! i.e 1 < 1 ⇒ not true P(1) ⇒ 2¹ < 1! i.e 2 < 1 ⇒ not true P(2) ⇒ 2² < 2! i.e 4 < 2 ⇒ not true P(3) ⇒ 2³ < 3! i.e 8 < 6 ⇒ not true P(4) ⇒ 2⁴ < 4! i.e 16 < 24 ⇒ true P(5) ⇒ 2⁵ < 5! i.e 32 < 60 ⇒ true, etc.

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»

According to the question,

P(n) which is true for all n ≥ 4 but P(1), P(2) and P(3) are not true

Let P(n) be 2ⁿ < n!

So, the examples of the given statements are,

P(0) ⇒ 2⁰ < 0!

i.e 1 < 1 ⇒ not true

P(1) ⇒ 2¹ < 1!

i.e 2 < 1 ⇒ not true

P(2) ⇒ 2² < 2!

i.e 4 < 2 ⇒ not true

P(3) ⇒ 2³ < 3!

i.e 8 < 6 ⇒ not true

P(4) ⇒ 2⁴ < 4!

i.e 16 < 24 ⇒ true

P(5) ⇒ 2⁵ < 5!

i.e 32 < 60 ⇒ true, etc.