Question

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 1

Answer:

An example of such statement may be:

P(n) : 2^n < n!

where n € N and n ≥ 4.

Here, P(1) , P(2) and P(3) are not true.

Justification:-

Case(1) : If n = 1

LHS = 2^1 = 2

RHS = 1! = 1

Case(2) : If n = 2

LHS = 2^2 = 4

RHS = 2! = 2

Case(3) : If n = 3

LHS = 2^3 = 8

RHS = 3! = 6.

Here, we can observe that , in all these three cases, LHS > RHS

ie, LHS is not less than RHS.

But the statement say that LHS must be less then RHS.

Hence, here P(1) , P(2) and P(3) are not true.

Now,

For n = 4

LHS = 2^4 = 16

RHS = 4! = 24

Here, LHS < RHS

Thus, P(4) is true.

For n = 5

LHS = 2^5 = 32

RHS = 5! = 120

Here, LHS < RHS

Thus, P(5) is true.

Similarly,

P(n) is true for every natural number which is greater than or equal to 4.

Thus,

If we consider an example of a statement such that;

P(n) : 2^n < n!

where n € N and n ≥ 4.

Then, P(1) , P(2) and P(3) are not true.