Bernoulli's inequality

Statement

Let $x$ be a real number with $x \ge -1$, and let $n$ be a natural number. Then $$(1 + x)^n \ge 1 + n x.$$

Remarks

When $x \neq 0$ and $n \ge 2$, the inequality is strict. For natural numbers $n$, the inequality also holds when $x \ge -2$. A proof by induction on $n$ multiplies $(1 + x)^n \ge 1 + n x$ by $1 + x \ge 0$ and uses $n x^2 \ge 0$.

Explanations

L1 human

A small growth rate $x$, applied $n$ times, gives at least $n$ times that growth. For example, $1.1^{10}$ is at least $1 + 10 \cdot 0.1 = 2$; its true value is about $2.59$. The difference comes from the growth on the growth, which the right side ignores.