Principle of Mathematical Induction Test

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Principle of Mathematical Induction Test - 6

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Weekly Quiz Competition

Question 1
1 / -0

Let P(n):n² − n + 41 is a prime number . then :

A
P ( 1 ) is not true

B
P ( 41 ) is not true

C
P ( 3) is not true

D
P ( 5 ) is not true

Solution

Since when n = 41 we have 〖41〗², which is not a prime number.
Question 2
1 / -0

Let that P(n) ⇒ P(n + 1) for all natural numbers n. also , if P ( m ) is true , m ∈ N , then we conclude that

A
P ( n ) is true for all n ≥ m

B
P ( n ) is true for all n < m

C
P ( n ) is true for all n

D
none of these.

Solution

This criteria is from the basic principle of mathematical induction.
Question 3
1 / -0

Consider the statement P ( n ) : “n² ≥ 100 “ . Here P(n) ⇒ P(n + 1) for all natural numbers n . Does it mean

A
p( n ) is true for all n

B
P ( n ) is true for all n ≥2

C
P ( n ) is true for all n ≥ 3

D
none of these

Solution

Since the statement is not true for n = 1 ,we cannot predict the validity for all natural numbers n.
Question 4
1 / -0

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

A
2

B
4

C
1

D
3

Solution

Since the values of n as 1,2,3gives inavlid inequations but when n = 4 we have 16 < 24, which is valid.
Question 5
1 / -0

If xⁿ−1 is divisible by x – k for all n belongs to natural numbers N , then the least positive integral value of k is :

A
1

B
3

C
4

D
2

Solution

since we have x - 1 as a factor of xⁿ - 1ⁿ.
Question 6
1 / -0

A student was asked to prove a statement P ( n ) by method of induction . He proved P ( k + 1 ) is true whenever P ( k ) Is true for all k ≥ 5 , k ∈ N and P ( 5 ) is true . On the basis of this he could conclude that P ( n ) is true

A
for all n > 5

B
for all n < 5

C
for all n ≥ 5

D
none of these

Solution

Yes because if it is true for n = k then it is true for all n ,which is the basic concept of mathematical induction.
Question 7
1 / -0

If a,b,c ∈ N,aⁿ + bⁿ is divisible by c , when n is odd but not when n is even , then the value of c is :

A
a - b

B
a³+ b³

C
a+b

D
none of these

Solution

Since a+b will be a factor of aⁿ +bⁿ.
Question 8
1 / -0

1.2.3 + 2.3.4 + 3.4.5 + ………..up to n terms is equal to 1/4 n ( n + 1 ) ( n + 2 ) ( n + 3 ) is true for

A
for all integers n

B
whole numbers n

C
all natural numbers n

D
none of these

Solution

By the method of mathematical induction the given statement is true for n = 1,2,3.....
Question 9
1 / -0

1 + 2 + 3 + .................. n = 1/2(n(n + 1)), is trure

A
only for n>2

B
for all natural numbers n

C
for n = 2 only

D
none of these

Solution

BY the process of mathematical induction , the given statement is true for all natural numbers n.

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Re-Attempt Weekly Quiz Competition

Principle of Mathematical Induction Test - 6

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Principle of Mathematical Induction Test - 6

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Question 1

P ( 1 ) is not true

P ( 41 ) is not true

P ( 3) is not true

P ( 5 ) is not true

Solution

Question 2

P ( n ) is true for all n ≥ m

P ( n ) is true for all n < m

P ( n ) is true for all n

none of these.

Solution

Question 3

p( n ) is true for all n

P ( n ) is true for all n ≥2

P ( n ) is true for all n ≥ 3

none of these

Solution

Question 4

2

4

1

3

Solution

Question 5

1

3

4

2

Solution

Question 6

for all n > 5

for all n < 5

for all n ≥ 5

none of these

Solution

Question 7

a - b

a3 + b3

a+b

none of these

Solution

Question 8

for all integers n

whole numbers n

all natural numbers n

none of these

Solution

Question 9

only for n>2

for all natural numbers n

for n = 2 only

none of these

Solution

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a³+ b³