Timeless Theorems of Mathematics/Binomial Theorem

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 $n\geq 0$.

Proof

Proposition: For any real numbers $a,b$ and $n\geq 0,$ $(a+b)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}a^{n-k}b^{k}$ Where ${\binom {n}{k}}={\frac {n!}{k!(n-k)!}}$

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 $n=2,(a+b)^{n}=(a+b)^{2}$ $=a^{2}+2ab+b^{2}$ $={\binom {2}{0}}a^{2}b^{0}+{\binom {2}{1}}a^{1}b^{1}+{\binom {2}{2}}a^{0}b^{2}$ $=\sum _{k=0}^{2}{\binom {2}{k}}a^{2-k}b^{k}$

For $n=3,(a+b)^{n}=(a+b)^{3}$ $=a^{3}+3a^{2}b+3ab^{2}+b^{3}$ $={\binom {3}{0}}a^{3}b^{0}+{\binom {3}{1}}a^{2}b^{1}+{\binom {3}{2}}a^{1}b^{2}+{\binom {3}{3}}a^{0}b^{3}$ $=\sum _{k=0}^{3}{\binom {3}{k}}a^{3-k}b^{k}$

Let's assume $(a+b)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}a^{n-k}b^{k}$ for some $a\geq 0.$ Now we just have to show that the equation holds true for $n+1.$

$(a+b)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}a^{n-k}b^{k}$

Or, $(a+b)^{n}(a+b)=(\sum _{k=0}^{n}{\binom {n}{k}}a^{n-k}b^{k})(a+b)$

$=({\binom {n}{0}}a^{n}b^{0}+{\binom {n}{1}}a^{n-1}b^{1}+{\binom {n}{2}}a^{n-2}b^{2}+...+{\binom {n}{n}}a^{0}b^{n})(a+b)$

$=a({\binom {n}{0}}a^{n}b^{0}+{\binom {n}{1}}a^{n-1}b^{1}+{\binom {n}{2}}a^{n-2}b^{2}+...+{\binom {n}{n}}a^{0}b^{n})+b({\binom {n}{0}}a^{n}b^{0}+{\binom {n}{1}}a^{n-1}b^{1}+{\binom {n}{2}}a^{n-2}b^{2}+...+{\binom {n}{n}}a^{0}b^{n})$

$=({\binom {n}{0}}a^{n+1}b^{0}+{\binom {n}{1}}a^{n}b^{1}+{\binom {n}{2}}a^{n-1}b^{2}+...+{\binom {n}{n}}ab^{n})+({\binom {n}{0}}a^{n}b^{1}+{\binom {n}{1}}a^{n-1}b^{2}+{\binom {n}{2}}a^{n-2}b^{3}+...+{\binom {n}{n}}a^{0}b^{n+1})$

$={\binom {n}{0}}a^{n+1}b^{0}+a^{n}b^{1}({\binom {n}{1}}+{\binom {n}{0}})+a^{n-1}b^{2}({\binom {n}{2}}+{\binom {n}{1}})+...+ab^{n}({\binom {n}{n}}+{\binom {n}{n-1}})+{\binom {n}{n}}a^{0}b^{n+1})$

$={\binom {n+1}{0}}a^{n+1}b^{0}+{\binom {n+1}{1}}a^{n}b^{1}+{\binom {n+1}{2}}a^{n-1}b^{2}+...+{\binom {n+1}{n}}ab^{n}+{\binom {n+1}{n+1}}a^{0}b^{n+1})$

∴$(a+b)^{n+1}=\sum _{k=0}^{n+1}{\binom {n+1}{k}}a^{n+1-k}b^{k}$