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# Binomial Theorem and its proof by Inductive method

## Introduction

Let us start with simple equation:

we all know that

(a+b)^{0 }= 1

and if we want to go to the next levels then

(a+b) ^{1} = a+b

(a+b)^{2 }= (a+b)*(a+b) = a^{2}+b^{2}+2ab

(a+b)^{3} = (a+b)*(a+b)*(a+b) = a^{3}+b^{3}+3a^{2}b+3ab^{2}

now, what if we want to expand (a+b) with higher powers?

It would become a very tedious work and for these kind of binomial expansions with higher powers, we have a theorem called **Binomial Theorem** which can expand any power for (a+b).

Binomial Theorem is a fundamental theorem in Algebra that is used to expand expressions of the form (a+b)^{n}.

**Definition and Inductive Proof** :

Binomial Theorem is defines as follows:

For any integer positive integer n

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