How do I start the inductive step?

Question

Let $x$ be any real number greater than -1. Prove that $(1 + x)^n\;\ge\;1+nx$ for every $n\ge0$ by induction.

The basis step is easy. I am struggling with starting the inductive step. Can you give me tip on how to start proving that $(1 + x)^{k+1}\;\ge\;1+(k+1)x$ ?

Answer 1

By definition, $(1 + x)^{k + 1} = (1 + x)^k(1 + x)$. Now use the induction hypothesis that $(1 + x)^k \geq 1 + kx$.

Answer 2

Multiply:

$\begin{align} (1 + x)^k &\ge 1 + k x \\ (1 + x)^{k + 1} &\ge (1 + x) \cdot (1 + k x) \\ &\vdots \end{align}$