Mathematical Induction Test

Result

Mathematical Induction Test - 1

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Question 1
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
nothing can be said

B
all n ∈ N

C
n > 2

D
n > 1

Solution

Because the statement is incomplete without the conclusion/ RHS
Question 2
1 / -0

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

A
r

B
( r + 1 ) !

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!.
Question 3
1 / -0

Which of the following numbers will divide n³ + 3n² + 5n + 3 for all n belonging to natural numbers?

A
2

B
4

C
3

D
7
Question 4
1 / -0

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

A
2

B
120

C
24

D
6

Solution

If n = 1 then the statement becomes 1x2x3x4= 24 : the consecutive natural numbers when substituted will be multiples of 24.
Question 5
1 / -0

(xⁿ−yⁿ)is divisible by ( x - y ) for

A
all n ∈ N

B
n > 1

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

Which of the following will be true for every x ∈ R?

A
3x > x

B
3^x > x

C
Both (1) and (2)

D
None of these
Question 7
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 < 3

B
all n > 3

C
all n

D
all n > 2

Solution

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

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Re-Attempt Weekly Quiz Competition

Mathematical Induction Test - 1

Result

Mathematical Induction Test - 1

Score

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SHARING IS CARING

Question 1

nothing can be said

all n ∈ N

n > 2

n > 1

Solution

Question 2

r

( r + 1 ) !

n+r

r !

Solution

Question 3

2

4

3

7

Question 4

2

120

24

6

Solution

Question 5

all n ∈ N

n > 1

n > 2

none of these.

Solution

Question 6

3x > x

3x > x

Both (1) and (2)

None of these

Question 7

all n < 3

all n > 3

all n

all n > 2

Solution

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3^x > x