Timeless Theorems of Mathematics/Binomial Theorem: Difference between revisions

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Template:Bookcat The Binomial Theorem is a fundamental theorem in algebra that provides a formula to expand powers of binomials. It allows us to easily expand expressions, (a+b)n, where a and b are real numbers or variables, and n≥0.

Proof

Proposition: For any real numbers a,b and n≥0, (a+b)n=∑k=0n(nk)an−kbk Where (nk)=n!k!(n−k)!

Proof (Mathematical Induction):

For n=0,(a+b)n=(a+b)0=1

For n=1,(a+b)n=(a+b)1=(a+b)


For n=2,(a+b)n=(a+b)2 =a2+2ab+b2 =(20)a2b0+(21)a1b1+(22)a0b2 =∑k=02(2k)a2−kbk


For n=3,(a+b)n=(a+b)3 =a3+3a2b+3ab2+b3 =(30)a3b0+(31)a2b1+(32)a1b2+(33)a0b3 =∑k=03(3k)a3−kbk


Let's assume (a+b)n=∑k=0n(nk)an−kbk for some a≥0. Now we just have to show that the equation holds true for n+1.


(a+b)n=∑k=0n(nk)an−kbk

Or, (a+b)n(a+b)=(∑k=0n(nk)an−kbk)(a+b)

=((n0)anb0+(n1)an−1b1+(n2)an−2b2+...+(nn)a0bn)(a+b)

=a((n0)anb0+(n1)an−1b1+(n2)an−2b2+...+(nn)a0bn)+b((n0)anb0+(n1)an−1b1+(n2)an−2b2+...+(nn)a0bn)

=((n0)an+1b0+(n1)anb1+(n2)an−1b2+...+(nn)abn)+((n0)anb1+(n1)an−1b2+(n2)an−2b3+...+(nn)a0bn+1)

=(n0)an+1b0+anb1((n1)+(n0))+an−1b2((n2)+(n1))+...+abn((nn)+(nn−1))+(nn)a0bn+1)

=(n+10)an+1b0+(n+11)anb1+(n+12)an−1b2+...+(n+1n)abn+(n+1n+1)a0bn+1)

∴(a+b)n+1=∑k=0n+1(n+1k)an+1−kbk

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