Mathematical Inductions and Binomial Theorem

Principle Of Mathematical Induction:
The principle of mathematical induction is stated as follows:
If a proposition or statement S(n) for each positive integer n is such that

1. S(1) is true i.e., S(n) is true for n = 1 
2. S(k + 1) is true whenever S(k) is true for any positive integer k, then S(n) is true for positive integers.

Important Note:
There is no integer n for which 3ⁿ is even.

Principle Of Extended Mathematical Induction:
Let i be an integer. If a formula or statement S(n) for n ≥ i is such that

1. S(i) is true
2. S(k + 1) is true whenever S(k) is true for any integer n ≥ i.

Then S(n) is true for all integers n ≥ i.

Binomial Expansion:
An algebraic expression consisting of two terms such as a + x, x - 2y, ax + b etc., is called a binomial or a binomial expression.
We know by actual multiplication that
(a + x)² = a²+ 2ax + x²                  ....................(i)
(a + x)³ = a³ + 3a²x + 3ax² + x³       ....................(ii)
The right sides of (i) and (ii) are called binomial expansions of the binomial a + x for the indices 2 and 3 respectively.
Binomial Theorem:
The rule or formula for expansion of a binomial raised to any positive integral power n is called the binomial theorem for positive integral index n. For any positive integer n,