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

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

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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.

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

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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.

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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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.

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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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.

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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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.

This criteria is from the basic principle of mathematical induction.

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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.

Since a+b will be a factor of an +bn.

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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.

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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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

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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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.

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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