how to prove by induction the $ (1+x)^{n}>1+nx+nx^2$

Question

Prove by induction the formula

$ (1+x)^{n}>1+nx+nx^2$ for $x>0$ real number and $n\ge 3$

my try : multiply both sides by $(1+x)$

gives $ (1+x)^{n+1}>1+(n+1)x+(2n+nx)x^2$

have I done something wrong or what I do next?

$(2n+nx)$ must turn into $(n+1)$

Answer 1

You are almost there. Note that $2n+x \geq 2n \geq n+1$.

Hence, the induction step becomes:

We have $$(1+x)^n > 1 + nx + nx^2$$ Multiplying by $1+x$, we obtain \begin{align} (1+x)^{n+1} & = (1+x)^n (1+x)\\ & > (1+nx+nx^2)(1+x)\\ & = 1+ nx + nx^2 + x + nx^2 + nx^3\\ & = 1+(n+1)x + (n+n)x^2 + nx^3\\ & \geq 1+(n+1)x + (n+n)x^2 & (\because nx^3 \geq 0)\\ & \geq 1+(n+1)x + (n+1)x^2 & (\because n \geq 1) \end{align} which gives us the induction step.