so I have this problem. It asks me to prove an expression by induction.

> 
> Let $n$ be a positive integer, and $\alpha$ any nonnegative real
> number. Prove by induction that$$(1+\alpha)^n\ge 1+n\alpha
>  +\frac{n(n-1)}{2}\alpha ^2$$
> 
> 
> 

So I let $G(n) = (1+\alpha)^n\ge 1+n\alpha +\frac{n(n-1)}{2}\alpha ^2$. I showed the initial case when $n=0$ and got $1=1$. I then assume $G(k)$ is true $G(k) = (1+\alpha)^k\ge 1+k\alpha +\frac{k(k-1)}{2}\alpha ^2$. I then try to prove $G(k+1)$ is also true. $G(n) = (1+\alpha)^{k+1}\ge 1+(k+1)\alpha +\frac{(k+1)k}{2}\alpha ^2$. However, this is where I am stuck. I'm not sure where to go from here. Thanks.