# Principle of Mathematical Induction - Online Test

Q1. Let P(n)P be a statement and let P(n)⇒P(n+1) for all natural numbers n , then what will the nature of P(n) ?
Explaination / Solution:

Since the statement is conditional statement ( if....then).

Q2. Let  be a statement  , where n is a natural number , then  is true for
Explaination / Solution:

Since for example n = 4 will give LHS as 16 and RHS as 4! = 1.2.3.4= 24

Q3. If x  -1 , then the statement  is true for
Explaination / Solution:

For example , if x = 0 and n > 1 , say n = 2 then the given inequality will become 12 > 1 , which is not true. Hence both the conditions on x and n are needed.

Q4. If P ( n ) = 2+4+6+………………..+2n , n  N , then P ( k ) = k ( k + 1 ) + 2  P ( k + 1 ) = ( k + 1 ) ( k +2 ) + 2 for all k  N . So we can conclude that P ( n ) = n ( n + 1 ) +2 for
Explaination / Solution:

Because the statement is incomplete without the conclusion/ RHS

Q5. The smallest +ve integer n , for which  $n!<$ holds is
Explaination / Solution:

when n = 2 ,  L H S  :  n ! = 2 ! = 2x1 = 2    RHS:

But if n = 1 , the inequation does not hold good.

Q6. is divisible by ( x - y ) for
Explaination / Solution:

Replacing n by 1,2,3... we get the expression with the factor ( x - y ).Hence it is divisible by ( x - y ).

Q7. The greatest positive integer , which divides n ( n + 1 ) ( n + 2 ) ( n + 3 ) for all n ∈N , is
Explaination / Solution:

If n = 1 then the statement becomes 1x2x3x4= 24 : the consecutive natural numbers when substituted will be multiples of 24.

Q8. Let P ( n ) denote the statement  is odd , It is seen that  , therefore P ( n ) is true for all
Explaination / Solution:

Since if n = 1 the answer is 2 which is even . also p ( 1 ) = p ( 2 ) is also not true since p(2 ) is 6 which is even.

Q9. The greatest positive integer, which divides , is
Explaination / Solution:

If n = 0 the given expression becomes 1.2.3.4........r = r! Also when n = 1 one more extra term will be there in the product  2.3.4........which is also divisible by r!.

Q10. The statement P ( n ) : “
“ is
Explaination / Solution:

Replace n = 1, 2 3 then LHS = RHS