# Principle of Mathematical Induction - Online Test

Q1. A student was asked to prove a statement P ( n ) by method of induction . He proved that P ( 3 ) is true and that P(n)⇒P(n+1) for all natural numbers n . On the basis of this the criteria applied on n for P ( n ) to be true
Explaination / Solution:

Since by the principle of mathematical induction if p( n ) is true for n = 3 then that will be the minimum point to be considered for the process.Also that P ( n ) ⇒P(n+1)

Q2.  is divisible by 64 for all
Explaination / Solution:

when n = 1 the value is 64. By induction process the consecutive replacement of n = 2,3,4....will be multiples of 64.

Q3. The inequality  is true
Explaination / Solution:

Since when n = 1 , we get the inequality as 1>1, which is not true. also for n = 2 , we get 2>2, which is false. Hence the given statement is true for n>2

Q4. The statement  is true for all
Explaination / Solution:

Since if n = 1 we have 3>4, which is not true . If n = 2 then 9>8, which is true.

Q5. The statement  is true for all
Explaination / Solution:

when n = 1 , , which is not true.When n = 2,  , which is false as well. Also when n = 3 we get .When n = 4 ,  Hence it is trure frm n = 4 .

Q6. The inequality  is true for :
Explaination / Solution:

When n = 1  we get , and when n = 2 we get ,. when n = 3 , which are inavlid inequations. Only when n = 4 we get , which is valid.

Q7. For each  is divisible by :
Explaination / Solution:

When n = 1 we have a + b.And the subsequent substitution of n as 2,3,... will result in the expression whose factor is a + b.

Q8.

For each n  N , n (n + 1 ) ( 2n + 1 ) is divisible by :

Explaination / Solution:

When n = 1 the value is 6 . The subsequent substitution will give the value as a multiple of 6.

Q9. For each n  N , is divisible by :
Explaination / Solution:

When n = 1 we have 391 which is divisible by 17.

Q10. For each n  N , is divisible by :
Explaination / Solution:

When n = 1 the value is 7.