It is often used to prove a statement depending upon a natural number n.

Type I:           If P (n) is a statement depending upon n, then to prove it by induction, we proceed as follows:

(i)         Verify the validity of P (n) for n = 1.

(ii)        Assume that P(n) is true for some positive integer m and then using it establish the validity of P(n) for n = m + 1.

Then, P (n) is true for each n ∈ N.

Binomial Theorem - Mathematical Induction And Its Applications

Type II: If P (n) is a statement depending upon n but beginning with some positive integer k, then to prove P (n), we proceed as follows:

(i)         Verify the validity of P (n) for n = k.

(ii)        Assume that the statement is true for n = m ≥ k. Then, using it establish the validity of P (n) for n = m + 1.

Then, P (n) is true for each n ≥ k

Binomial Theorem - Mathematical Induction And Its Applications

Note:  1. Product of r consecutive integers is divisible by r !.

2. For x ≠ y, xn – yn is divisible by

(i) x + y if n is even

(ii) x – y if n is even or odd.

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CBSE Class 11 Maths Binomial Theorem All Topic Notes CBSE Class 11 Maths All Chapters Notes

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