Topic: Principle of Mathematical Induction (Test 1)



Topic: Principle of Mathematical Induction
Q.1
Let  be a statement  , where n is a natural number , then  is true for
A. all n > 2
B. all n < 3
C. all n > 3
D. all n
Answer : Option C
Explaination / Solution:

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

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Q.2
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
Answer : Option C
Explaination / Solution:

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

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Q.3
 is divisible by 64 for all
A. n ∈ N
B. n ∈ N , n ≥2
C. None of these
D. n ∈ N , n >2
Answer : Option A
Explaination / Solution:

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

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Q.4
For each  is divisible by :
A. (a+b)3
B. a + b
C. None of these
D. (a+b)2
Answer : Option B
Explaination / 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.

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Q.5
The statement P ( n ) : “ “ is true for :
A. all n ≥ 3
B. all n .
C. all n ≥ 2
D. no n ∈ N ,
Answer : Option D
Explaination / 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.

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Q.6
Let that  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
B. P ( n ) is true for all n < m
C. None of these
D. P ( n ) is true for all n≥ m
Answer : Option D
Explaination / Solution:

This criteria is from the basic principle of mathematical induction.

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Q.7
If  is divisible by c , when n is odd but not when n is even , then the value of c is :
A. a3+b3
B. none of these
C. a+b
D. a-b
Answer : Option C
Explaination / Solution:

Since a+b will be a factor of an +bn.

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Q.8
If  is divisible by 64 for all n  N , then the least negative integral value of  is
A. -4
B. -2
C. -1
D. -3
Answer : Option C
Explaination / Solution:

When n = 1 we have the value of the expression as 65 . Given that the expression is divisible be 64. Hence the value is -1.

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Q.9
up to n terms is equal to
A. n3(2n+3)
B. n(2n+3)
C. None of these
D. 1(n+2)(n+4)
Answer : Option A
Explaination / Solution:

By the process of mathematical induction when n = 1 we have 1/15. When n = 2 we have LHS : 1/15+1/35=2/21, RHS :2/(3(4+3))=2/21, which is true

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Q.10
The smallest positive integer for which The statement  is true for
A. 2
B. 3
C. 4
D. 1
Answer : Option C
Explaination / 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.

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