Principle of Mathematical Induction - Online Test
Q1. 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
Answer : Option D
Explaination / 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.
Q2. If is divisible by c , when n is odd but not when n is even , then the value of c is :
Answer : Option C
Explaination / Solution:
Since a+b will be a factor of an +bn.
Q3. 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
Answer : Option A
Explaination / Solution:
By the method of mathematical induction the given statement is true for n = 1,2,3.....
Q4. 1 + 2 + 3 + .................. n = , is trure
Answer : Option A
Explaination / Solution:
BY the process of mathematical induction , the given statement is true for all natural numbers n.
Q5. The sum of the terms in the nth bracket of the series 1 + (2+3+4) + (5+6+7+8+9) ….is
Answer : Option D
Explaination / Solution:
replace n = 1 we get the first term and n = 2 we get the sum of first two terms....
Q6. For all is divisible by :
Answer : Option C
Explaination / Solution:
Replace n = 1 we get 209 which is divisible by 11.
Q7. If is divisible by 64 for all n N , then the least negative integral value of is
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.
Q8. to n terms is equal to
Answer : Option D
Explaination / Solution:
When n = 1 we get 3/4, and the subsequent terms when n is replaced by 2,3,4...
Q9. is divisible by :
Answer : Option D
Explaination / Solution:
put n = 1 we get 11 .
Q10. For all odd positive integers n , the number is divisible by :
Answer : Option B
Explaination / Solution:
When n = 1 we have 0 : When n = 2 we have 24. When n = 3 we have 120 .....where all these values are multiples of 24.
