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 \(n \lt m^n.\)