Principle of Mathematical Induction Test

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Principle of Mathematical Induction Test - 3

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Question 1
1 / -0

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

A
for all n ≥ 3

B
for all n ∈ N

C
for no n

D
none of these

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)
Question 2
1 / -0

3²ⁿ⁺²− 8n − 9 is divisible by 64 for all

A
n ∈ N

B
n ∈ N , n > 2

C
n ∈ N , n ≥2

D
none of these.

Solution

when n = 1 the value is 64. By induction process the consecutive replacement of n = 2,3,4....will be multiples of 64.
Question 3
1 / -0

The inequality n! > 2ⁿ⁻¹ is true

A
for no n ∈ N

B
for all n ∈ N

C
for all n > 1

D
for all n> 2 ,n ∈ N

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
Question 4
1 / -0

The statement 3ⁿ > 4n is true for all

A
n ∈ N

B
n > 1 , n ∈ N

C
n > 2 , n ∈ N

D
none of these

Solution

Since if n = 1 we have 3 > 4, which is not true . If n = 2 then 9 > 8, which is true.
Question 5
1 / -0

The statement 2ⁿ> 3n is true for all

A
n > 3

B
n > 4

C
n ∈ N

D
none of these.

Solution

when n = 1 , 2>3, which is not true.When n = 2, 4>6, which is false as well. Also when n = 3 we get 8>9.When n = 4 , 16>12 Hence it is trure frm n = 4 .
Question 6
1 / -0

The statement 2ⁿ⁺² < 3ⁿ is true for all

A
for all n ∈ N

B
all n > 3

C
all n > 2

D
none of these.

Solution

When n = 1 8<3, is not true. When n = 2 16<9, is false. When n = 3 32 < 27, is not valid. But when n = 4 64 < 81, is true.
Question 7
1 / -0

The smallest positive integer for which The statement 3ⁿ + 1 < 4ⁿ is true for

A
1

B
2

C
3

D
4

Solution

When n = 1 9<4 not valid. when n = 2 27<16 not true. when n = 3 81<64 is in correct. when n = 4 243 < 256 , is true.
Question 8
1 / -0

For all n ∈∈ N , 49ⁿ + 16n − 1is divisible by

A
3

B
19

C
64

D
29

Solution

Replace n = 1 , we have 64.
Question 9
1 / -0

2³ⁿ-1 is divisible by

A
6

B
7

C
3

D
none of these

Solution

If n = 1 we get 7. n = 2 we get 63 which is divisible by 7........

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Re-Attempt Weekly Quiz Competition

Principle of Mathematical Induction Test - 3

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Principle of Mathematical Induction Test - 3

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Question 1

for all n ≥ 3

for all n ∈ N

for no n

none of these

Solution

Question 2

n ∈ N

n ∈ N , n > 2

n ∈ N , n ≥2

none of these.

Solution

Question 3

for no n ∈ N

for all n ∈ N

for all n > 1

for all n> 2 ,n ∈ N

Solution

Question 4

n ∈ N

n > 1 , n ∈ N

n > 2 , n ∈ N

none of these

Solution

Question 5

n > 3

n > 4

n ∈ N

none of these.

Solution

Question 6

for all n ∈ N

all n > 3

all n > 2

none of these.

Solution

Question 7

1

2

3

4

Solution

Question 8

3

19

64

29

Solution

Question 9

6

7

3

none of these

Solution

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