Binomial Theorem

Theorem
Let $$x, y \in \mathbb{R}$$.

Then $$\forall n \in \mathbb{Z}_+: \left({x+y}\right)^n = \sum_{k=0}^n {n\choose k}x^{n-k}y^k$$.

Proof
Base Case:

$$n=0$$

$$(x+y)^0=1={0\choose 0}x^{0-0}y^0=\sum_{k=0}^0 {0\choose k}x^{0-k}y^k$$

Therefore the base case holds.

Inductive Hypothesis:

$$(x+y)^j = \sum_{k=0}^j {j\choose k}x^{j-k}y^k$$ for all $$j \ge 1$$

Inductive Step:

And so we are done by the Principle of Mathematical Induction.