Question

If P(n) : (5n)/2 is less than n!, n belongs to Natural numbers, then P(n) is true for n
1.  greater than 2
2. All greater than 3
3. All less than 4
4. None of these

Answer 1

Correct Answer - Option 2 : All greater than 3

Given:

P(n) : 5n/2 < n!

n belongs to Natural number

Calculation:

P(n) : (5n)/2 < n!

Put n = 1, 2, 3 ,4, ......

For n = 1

P(1) : 5(1)/2  > 1! = 2 > 1 (false)

For n = 2

P(2) : 5(2)/2  > 2! = 5 > 2 (false)

For n = 3

P(3) : 5(3)/2  > 3! = 7.5 > 6 (false)

For n = 4

P(4) : 5(4)/2  > 4! = 10 < 24 (True)

similarly for P(5) ,P(6).... and for all n natural numbers it is true

∴ P(n) is true for n all greater than 3