Mathematical Induction Test

Result

Mathematical Induction Test - 2

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Question 1
1 / -0

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

A
it satisfies only all n > 1

B
Nature cannot be identified

C
It is true for all n > m , m being a fixed positive integer

D
true for all n

Solution

Since the statement is conditional statement ( if....then).
Question 2
1 / -0

The smallest +ve integer n , for which n!<(n+1/2)ⁿholds is

A
4

B
2

C
1

D
3

Solution

when n = 2 , L H S : n ! = 2 ! = 2x1 = 2 RHS: (3/2)2=9/4

But if n = 1 , the inequation does not hold good.
Question 3
1 / -0

Let P ( n ) denote the statement n²+n is odd , It is seen that P(n)⇒P(n+1) , therefore P ( n ) is true for all

A
n > 2

B
n

C
n > 1

D
none of these

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
Question 4
1 / -0

The statement P ( n ) : “1 X 1! + 2 X 2! + 3 X 3! + …..+ n X n! = ( n + 1 ) !− 1 “ is

A
true for all n > 1

B
true for all n ∈ N

C
not true for any n

D
none of these

Solution

Replace n = 1, 2 3 then LHS = RHS