Principle of Mathematical Induction - Online Test
Q1. A student was asked to prove a statement P ( n ) by method of induction . He proved that P ( 3 ) is true and that P(n)⇒P(n+1) for all natural numbers n . On the basis of this the criteria applied on n for P ( n ) to be true
Answer : Option A
Explaination / Solution:
Since by the principle of mathematical induction if p( n ) is true for n = 3 then that will be the minimum point to be considered for the process.Also that P ( n ) ⇒P(n+1)
Q2. is divisible by 64 for all
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.
Q3. The inequality is true
Answer : Option C
Explaination / Solution:
Since when n = 1 , we get the inequality as 1>1, which is not true. also for n = 2 , we get 2>2, which is false. Hence the given statement is true for n>2
Q4. The statement is true for all
Answer : Option C
Explaination / Solution:
Since if n = 1 we have 3>4, which is not true . If n = 2 then 9>8, which is true.
Q5. The statement is true for all
Answer : Option C
Explaination / Solution:
when n = 1 , , which is not true.When n = 2, , which is false as well. Also when n = 3 we get .When n = 4 , Hence it is trure frm n = 4 .
Q6. The inequality is true for :
Answer : Option B
Explaination / Solution:
When n = 1 we get , and when n = 2 we get ,. when n = 3 , which are inavlid inequations. Only when n = 4 we get , which is valid.
Q7. For each is divisible by :
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.
Q8.
For each n N , n (n + 1 ) ( 2n + 1 ) is divisible by :
Answer : Option A
Explaination / Solution:
When n = 1 the value is 6 . The subsequent substitution will give the value as a multiple of 6.
Q9. For each n N , is divisible by :
Answer : Option C
Explaination / Solution:
When n = 1 we have 391 which is divisible by 17.
Q10. For each n N , is divisible by :
Answer : Option A
Explaination / Solution:
When n = 1 the value is 7.
