<s>[INST] mathematical induction ($(1+x)^n\ge1+nx+n(n-1)x^2/2$) Suppose that $x > 0$ and let $n \geq 2$ be a positive integer. Prove that $(1 + x)^n \geq 1 + nx + \frac{n(n-1)}{2}x^2$
So for the base case, I have $x=1$, but that really is not getting me anywhere.
Would it be better to manipulate $n$?  Is there a way to know which one to manipulate?
 [/INST] HINT:
Let $(1+x)^m\ge \{1+mx+\frac{m(m-1)}2x^2\}$
So, $(1+x)^{m+1}=(1+x)(1+x)^m \ge (1+x)\{1+mx+\frac{m(m-1)}2x^2\}$
$=1+(m+1)x+x^2\{\frac{m(m-1)}2+m\}+x^3\frac{m(m-1)}2\ge \{1+(m+1)x+\frac{m(m+1)}2x^2\}$  as $x>0$ and $m\ge2$
Now, for $m=2,1+2x+x^2-\{1+2x+x^2\}=0\implies (1+x)^2\ge \{1+2x+\frac{2(2-1)}2x^2\}$
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