Principle of Mathematical Induction Test

Result

Principle of Mathematical Induction Test - 5

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Weekly Quiz Competition

Question 1
1 / -0

The inequality 2ⁿ + 1 < n! is true for :

A
all n > 3

B
all n > 1

C
all n

D
none of these

Solution

When n = 1 we get 3 < 1, and when n = 2 we get 5 < 2,. when n = 3 9 < 6, which are inavlid inequations. Only when n = 4 we get 17 < 24, which is valid.
Question 2
1 / -0

For each n ∈ N,a²ⁿ⁻¹ + b²ⁿ ^{− 1} is divisible by :

A
(a+b)²

B
(a+b)³

C
a + b

D
none of these.

Solution

When n = 1 we have a + b.And the subsequent substitution of n as 2,3,... will result in the expression whose factor is a + b.
Question 3
1 / -0

For each n ∈ N , n (n + 1 ) ( 2n + 1 ) is divisible by :

A
2

B
6

C
15

D
8

Solution

When n = 1 the value is 6 . The subsequent substitution will give the value as a multiple of 6.
Question 4
1 / -0

For each n ∈ N , 3(5²ⁿ⁺¹) + 23ⁿ⁺¹ is divisible by :

A
19

B
21

C
17

D
23

Solution

When n = 1 we have 391 which is divisible by 17.
Question 5
1 / -0

For each n ∈ N , 2³ⁿ⁻¹ is divisible by :

A
16

B
7

C
8

D
none of these.

Solution

When n = 1 the value is 7.
Question 6
1 / -0

For each n ∈ N , 3²ⁿ⁻¹ is divisible by :

A
32

B
8

C
16

D
19

Solution

when n = 1 the value is 8 which is the multiple pf 8.
Question 7
1 / -0

The statement P ( n ) : “(n + 3)² > 2ⁿ⁺³ “ is true for :

A
no n ∈ N ,

B
all n .

C
all n ≥ 2

D
all n ≥ 3

Solution

When n = 1 we get 16 > 16, which is false. when n = 2 we get 25 > 32,which is false as well. As n = 3,4,5....the inequalty does not hold correct.
Question 8
1 / -0

The smallest positive integer ‘ n ‘ for which 2ⁿ(1×2×3×...............×n) < nⁿ holds is :

A
5

B
3

C
6

D
none of these

Solution

When n = 1 ,we get 2 < 1,which is not valid. When n takes values 2,3,4,5 the inequality is invalid . But when n = 6 we have 46080 < 46656,which is valid.
Question 9
1 / -0

If 10ⁿ + 3 × 4ⁿ⁺¹ + k is divisible by 9 for all n ∈ N , then the least positive integral value of k is :

A
3

B
1

C
7

D
5

Solution

When n = 1 we have 58. Given that the expression is divisible by 9 . Hence the least value of k will be 5,so that the value becomes 63 which is divisible by 9.

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Re-Attempt Weekly Quiz Competition

Principle of Mathematical Induction Test - 5

Result

Principle of Mathematical Induction Test - 5

Score

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SHARING IS CARING

Question 1

all n > 3

all n > 1

all n

none of these

Solution

Question 2

(a+b)2

(a+b)3

a + b

none of these.

Solution

Question 3

2

6

15

8

Solution

Question 4

19

21

17

23

Solution

Question 5

16

7

8

none of these.

Solution

Question 6

32

8

16

19

Solution

Question 7

no n ∈ N ,

all n .

all n ≥ 2

all n ≥ 3

Solution

Question 8

5

3

6

none of these

Solution

Question 9

3

1

7

5

Solution

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(a+b)²

(a+b)³