<s>[INST] How do I prove that $(1+\frac{1}{2})^{n} \ge 1 + \frac{n}{2}$ for every $n \ge 1$? How do I prove that $(1+\frac{1}{2})^{n} \ge 1 + \frac{n}{2}$ for every $n \ge 1$
My base case is $n=1$ 
Inductive step is $n=k$ 
Assume $n=k+1$
$(\frac{3}{2})^{k} \times \frac{3}{2} \ge (1 + \frac{k+1}{2})$
I'm not sure how to proceed.
 [/INST] Let $P(n):(1+x)^n\ge 1+nx$
Clearly $P(n)$ is true for $n=1$
Let $P(n)$ is true for $n=m\implies (1+x)^m\ge 1+mx$
$\implies (1+x)^{m+1}\ge (1+mx)(1+x)=1+(m+1)x+mx^2\ge 1+(m+1)x$ if $m\ge0$  and $x$ is real
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