Binomial Theorem Tutorial, Series Expansion Formula, Example, Proof

The binomial theorem describes the algebraic expansion of powers of a binomial. This theorem is a quick way of expanding a binomial expression that has been raised to some power.

Binomial Theorem Tutorial Proof, Example

Statement
If n is a natural number, then the series expansion formula is as follows,
(x+a)n=nC0xn+nC1xn-1a+nC2xn-2a2+.........+nCrxn-rar+............+nCnan

Proof:
Let P(n) = (x+a)n=nC0xn+nC1xn-1a+nC2xn-2a2+.........+nCrxn-rar+............+ nCnan--------(1)

Consider n=1 in the above statement,
Then, LHS of P(1)=x+a
RHS of P(1)=1 * x +1 * a =x+a=LHS of P(1)
Therefore,P(1) is true

Let us assume that the statement P(k) be true for k belongs to N where N is a set of natural numbers
Now, substitute the value 'k' instead of 'n' in the equation,
(x+a)k=kC0xk+kC1xk-1a+kC2xk-2a2+.........+kCrxk-rar+...........+kCkak is true-------(2)

To prove P(k+1) is true
Substitute the value 'k+1' instead of 'n' in the equation (1),
(x+a)k+1=(k+1)C0xk+1+(k+1)C1xka+(k+1)C2xk-1a2+.........+
(k+1)Crxk+1-rar+...........+(k+1)C(k+1)ak+1 is true


(x+a)k+1=(x+a)(x+a)k
From the equation (2),
(x+a)k+1=(x+a)[kC0xk+kC1xk-1a+kC2xk-2a2+.........+kCrxk-rar+...........+kCkak]
=kC0xk+1+kC1xka+kC2xk-1a2+......+kCrxk+1-rar +.........+kCkxak+kC0xka+kC1xk-1a2+.....+ KC(r-1)xk+1-rar+......+kCkak+1
=kC0xk+1+(kC1+kC0)xka+........+(kCr+kC(r-1))xk+1-rar+....+kCka(k+1)


But according to the binomial formula, kCr+kC(r-1)=(k+1)Cr
Put r=1,2,... in the above formula,
kC1+kC0=(k+1)C1
kC1+kC0=(k+1)C2......
kC0=1=(k+1)C0;
kCk=1=(k+1)C(k+1);

(x+a)k+1=(k+1)C0xk+1+(k+1)C1xka+(k+1)C2xk-1a2+.........+(k+1)Crxk+1-rar+...........+(k+1)C(k+1)ak+1

Therefore P(k+1) is true.

Thus if P(k) is true,then P(k+1) is also true.

Therefore by the principle of Mathematical induction, P(n) is true for all n belongs to N.

Where N is the set of natural numbers.

Thus the binomial series expansion theorem is proved with example.
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