First Principle of Mathematical Induction
Category : JEE Main & Advanced
The proof of proposition by mathematical induction consists of the following three steps :
Step I : (Verification step) : Actual verification of the proposition for the starting value \[''i''\].
Step II : (Induction step) : Assuming the proposition to be true for \[''k'',\,\ge i\] and proving that it is true for the value \[(k+1)\] which is next higher integer.
Step III : (Generalization step) : To combine the above two steps. Let \[p(n)\] be a statement involving the natural number n such that
(i) \[p(1)\] is true i.e. \[p(n)\] is true for \[n=1\].
(ii) \[p(m+1)\] is true, whenever \[p(m)\] is true i.e. \[p(m)\] is true
\[\Rightarrow \] \[p(m+1)\] is true.
Then \[p(n)\] is true for all natural numbers \[n\].
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