# Elementary number theory

If  $$x \ne 1,$$  then  $$\displaystyle \sum_{k = 0}^{n - 1} x^k = \dfrac{1 - x^n}{1 - x}.$$
Induction Proof for Sum of First N Powers of Real Numbers | Number Theory, Proofs
Wrath of Math
$$\displaystyle \sum_{k = 0}^{n - 1} x^k = \dfrac{1 - x^n}{1 - x}$$
The theorem below relies on the theorem above.
If  $$n, m \in \mathbb{Z}^+$$  and  $$m \gt 1,$$  then  $$n \lt m^n.$$
Proof: N is less than M to the Power of N | Number Theory
Wrath of Math

Let $$n, m \in \mathbb{Z}^+.$$ If $$m \gt 1,$$ then $$n \lt m^n.$$