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  • To prove a result by mathematical induction what values do we need to check for n?Option: 1 n=1,n=kOpti

To prove a result by mathematical induction what values do we need to check for n?

Option: 1

n=1,n=k


Option: 2

n=1,n=k,n=k+1


Option: 3

n=1,n=k-1,n=k


Option: 4

Both (B) and (C)


Answers (1)

Principle of Mathematical Induction

Suppose there is a given statement P(n) involving the natural number n such that

  1. The statement is true for n = 1, i.e., P(1) is true, and  

  2. If the statement is true for n = k (where k is some positive integer), then the statement is also true for n = k + 1, i.e., truth of P(k) implies the truth of P (k + 1). 

Then, P(n) is true for all natural numbers n.

Property (ii) is a conditional property. It does not assert that the given statement is true for n = k, but only that if it is true for n = k, then it is also true for n = k +1. So, to prove that the property holds, only prove that conditional proposition: 

If the statement is true for n = k, then it is also true for n = k + 1.

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As long as the terms are consecutive like (k, k+1) or (k-1,k), Principle of Mathematical Induction can be applied

So, both B and C options are correct

Posted by

Kshitij

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