Timeless Theorems of Mathematics/Binomial Theorem

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 $n \geq 0$ .

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

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

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

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

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

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

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

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

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

Timeless Theorems of Mathematics/Binomial Theorem

Proof

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