Principle of Mathematical Induction Test

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

C
true for all n

D
Nature cannot be identified.

Solution

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

Let P(n) b e a statement 2n < n! , where n is a natural number , then P(n)is true for

A
all n

B
all n < 3

C
all n > 2

D
all n > 3

Solution

Since for example n = 4 will give LHS as 16 and RHS as 4! = 1.2.3.4= 24
Question 3
1 / -0

If x > -1 , then the statement ( 1 + x ) n > 1 + nx is true for

A
all n > 1 provided x ≠ 0

B
all n > 1

C
all n ∈ N

D
none of these

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.
Question 4
1 / -0

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

A
n > 2

B
n > 1

C
all n ∈ N

D
nothing can be said

Solution

Because the statement is incomplete without the conclusion/ RHS
Question 5
1 / -0

(xⁿ − yⁿ) is divisible by ( x - y ) for

A
n > 1

B
all n ∈ N

C
n > 2

D
none of these.

Solution

Replacing n by 1,2,3... we get the expression with the factor ( x - y ).Hence it is divisible by ( x - y ).
Question 6
1 / -0

The greatest positive integer , which divides n ( n + 1 ) ( n + 2 ) ( n + 3 ) for all n ∈ N , is

A
2

B
6

C
24

D
120

Solution

If n = 1 then the statement becomes 1x2x3x4= 24 : the consecutive natural numbers when substituted will be multiples of 24.
Question 7
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 > 1

C
n

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 8
1 / -0

The greatest positive integer, which divides (n + 1) (n + 2) (n + 3)..................(n + r)∀n ∈ W , is

A
( r + 1 ) !

B
r !

C
n + r

D
r

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........(r+1)which is also divisible by r!.

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Re-Attempt Weekly Quiz Competition

Principle of Mathematical Induction Test - 2

Result

Principle of Mathematical Induction Test - 2

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SHARING IS CARING

Question 1

it satisfies only all n > 1

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

true for all n

Nature cannot be identified.

Solution

Question 2

all n

all n < 3

all n > 2

all n > 3

Solution

Question 3

all n > 1 provided x ≠ 0

all n > 1

all n ∈ N

none of these

Solution

Question 4

n > 2

n > 1

all n ∈ N

nothing can be said

Solution

Question 5

n > 1

all n ∈ N

n > 2

none of these.

Solution

Question 6

2

6

24

120

Solution

Question 7

n > 2

n > 1

n

none of these

Solution

Question 8

( r + 1 ) !

r !

n + r

r

Solution

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