# Principle of Mathematical Induction - Online Test

Q1. For each n  N ,  is divisible by :
Explaination / Solution:

when n = 1 the value is 8 which is the multiple pf 8.

Q2. The statement P ( n ) : “ “ is true for :
Explaination / Solution:

When n = 1 we get 16>16, which is false. when n = 2 we get 25>32,which is false as well. As n = 3,4,5....the inequalty does not hold correct.

Q3.

The smallest positive integer ‘ n ‘ for which holds is :

Explaination / Solution:

When n = 1 ,we get ,which is not valid. When n takes values 2,3,4,5 the inequality is invalid . But when n = 6 we have ,which is valid.

Q4.

If is divisible by 9 for all n  N , then the least positive integral value of k is :

Explaination / Solution:

When n = 1 we have 58. Given that the expression is divisible by 9 . Hence the least value of k will be 5,so that the value becomes 63 which is divisible by 9.

Q5. The smallest positive integer n , for which  holds :
Explaination / Solution:

When n = 1 we get ,which is invalid. When n = 2 we get, which is valid.

Q6.

Let  is a prime number . then :

Explaination / Solution:

Since when n = 41 we have , which is not a prime number.
Q7. Let that  for all natural numbers n. also , if P ( m ) is true , m  N , then we conclude that
Explaination / Solution:

This criteria is from the basic principle of mathematical induction.

Q8. Consider the statement P ( n ) : “ “ . Here for all natural numbers n . Does it mean
Explaination / Solution:

Since the statement is not true for n = 1 ,we cannot predict the validity for all natural numbers n.

Q9. The smallest positive integer ‘ n ‘ for which P ( n ) : holds is :
Explaination / Solution:

Since the values of n as 1,2,3gives inavlid inequations but when n = 4 we have 16<24, which is valid.

Q10.

If  is divisible by x – k for all n belongs to natural numbers N , then the least positive integral value of k is :

Explaination / Solution:

since we have x - 1 as a factor of xn - 1n.