JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Second Principle of Mathematical Induction

The proof of proposition by mathematical induction consists of following steps :

Step I : (Verification step) : Actual verification of the proposition for the starting value \[i\] and \[(i+1)\].

Step II : (Induction step) : Assuming the proposition to be true for \[k-1\] and \[k\] and then proving that it is true for the value \[k+1;\,\,k\,\,\ge \,\,i+1\].

Step III : (Generalization step) : Combining 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 \[n=1\] and

(ii) \[p(m+1)\] is true, whenever \[p(n)\] is true for all \[n,\] where \[i\le n\le m\].

Then \[p(n)\]is true for all natural numbers.

For \[a\ne b,\] The expression \[{{a}^{n}}-{{b}^{n}}\] is divisible by

(a) \[a+b,\] if \[n\] is even.                   

(b) \[a-b,\] if \[n\] is odd or even.